Embedded newform 170.3.p.a.11.6

• Label: 170.3.p.a.11.6
• Level: $170$
• Weight: $3$
• Character: 170.11
• 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			11.6
Character	\(\chi\)	\(=\)	170.11
Dual form			170.3.p.a.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30656 - 0.541196i) q^{2} +(0.718320 + 3.61124i) q^{3} +(1.41421 + 1.41421i) q^{4} +(1.85922 + 1.24229i) q^{5} +(1.01586 - 5.10706i) q^{6} +(7.14141 - 4.77174i) q^{7} +(-1.08239 - 2.61313i) q^{8} +(-4.21015 + 1.74390i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{3} + 16 q^{6} - 16 q^{7} + 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(71\)	\(137\)
\(\chi(n)\)	\(e\left(\frac{7}{16}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30656	−	0.541196i	−0.653281	−	0.270598i
							\(3\)	0.718320	+	3.61124i	0.239440	+	1.20375i	0.894115	+	0.447837i	\(0.147805\pi\)
−0.654675	+	0.755910i	\(0.727195\pi\)
\(5\)	1.85922	+	1.24229i	0.371845	+	0.248459i
							\(7\)	7.14141	−	4.77174i	1.02020	−	0.681677i	0.0713667	−	0.997450i	\(-0.477264\pi\)
0.948835	+	0.315773i	\(0.102264\pi\)
\(11\)	12.3658	+	2.45972i	1.12417	+	0.223611i	0.721958	−	0.691937i	\(-0.243242\pi\)
							0.402209	+	0.915548i	\(0.368242\pi\)
\(13\)	−8.64640	+	8.64640i	−0.665108	+	0.665108i	−0.956579	−	0.291472i	\(-0.905855\pi\)
							0.291472	+	0.956579i	\(0.405855\pi\)
\(17\)	−0.924698	+	16.9748i	−0.0543940	+	0.998520i
							\(19\)	−12.5974	−	5.21802i	−0.663022	−	0.274633i	0.0256878	−	0.999670i	\(-0.491822\pi\)
−0.688709	+	0.725037i	\(0.741822\pi\)
\(23\)	−2.35902	+	11.8596i	−0.102566	+	0.515635i	0.895010	+	0.446047i	\(0.147169\pi\)
							−0.997576	+	0.0695883i	\(0.977831\pi\)
\(29\)	15.9370	−	23.8514i	0.549553	−	0.822464i	−0.447879	−	0.894094i	\(-0.647820\pi\)
							0.997431	+	0.0716307i	\(0.0228203\pi\)
\(31\)	−11.5281	+	2.29309i	−0.371875	+	0.0739706i	−0.377490	−	0.926014i	\(-0.623213\pi\)
							0.00561479	+	0.999984i	\(0.498213\pi\)
\(37\)	−6.46612	−	32.5074i	−0.174760	−	0.878577i	−0.964287	−	0.264861i	\(-0.914674\pi\)
							0.789527	−	0.613716i	\(-0.210326\pi\)
\(41\)	48.5146	−	32.4164i	1.18328	−	0.790644i	0.201284	−	0.979533i	\(-0.435489\pi\)
							0.981999	+	0.188889i	\(0.0604885\pi\)
\(43\)	−39.0158	+	16.1609i	−0.907345	+	0.375835i	−0.787040	−	0.616902i	\(-0.788387\pi\)
							−0.120305	+	0.992737i	\(0.538387\pi\)
\(47\)	−35.2697	+	35.2697i	−0.750419	+	0.750419i	−0.974557	−	0.224138i	\(-0.928043\pi\)
							0.224138	+	0.974557i	\(0.428043\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.a.11.6	✓	48
17.14	odd	16	inner	170.3.p.a.31.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.a.11.6	✓	48	1.1	even	1	trivial
170.3.p.a.31.6	yes	48	17.14	odd	16	inner