Embedded newform 170.3.p.a.11.1

• Label: 170.3.p.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	170.11
Dual form			170.3.p.a.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30656 - 0.541196i) q^{2} +(-1.03104 - 5.18337i) q^{3} +(1.41421 + 1.41421i) q^{4} +(-1.85922 - 1.24229i) q^{5} +(-1.45810 + 7.33039i) q^{6} +(-9.97529 + 6.66528i) q^{7} +(-1.08239 - 2.61313i) q^{8} +(-17.4893 + 7.24432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{3} + 16 q^{6} - 16 q^{7} + 32 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.03104	−	5.18337i	−0.343679	−	1.72779i	−0.636195	−	0.771528i	\(-0.719493\pi\)
0.292517	−	0.956260i	\(-0.405507\pi\)
\(5\)	−1.85922	−	1.24229i	−0.371845	−	0.248459i
							\(7\)	−9.97529	+	6.66528i	−1.42504	+	0.952182i	−0.426172	+	0.904642i	\(0.640138\pi\)
−0.998869	+	0.0475405i	\(0.984862\pi\)
\(11\)	14.2605	+	2.83658i	1.29640	+	0.257871i	0.794612	−	0.607117i	\(-0.207674\pi\)
							0.501792	+	0.864988i	\(0.332674\pi\)
\(13\)	1.63749	−	1.63749i	0.125961	−	0.125961i	−0.641316	−	0.767277i	\(-0.721611\pi\)
							0.767277	+	0.641316i	\(0.221611\pi\)
\(17\)	−7.78744	−	15.1114i	−0.458085	−	0.888908i
							\(19\)	−1.59351	−	0.660054i	−0.0838690	−	0.0347397i	0.340355	−	0.940297i	\(-0.389453\pi\)
−0.424224	+	0.905558i	\(0.639453\pi\)
\(23\)	−3.73526	+	18.7784i	−0.162403	+	0.816454i	0.810589	+	0.585615i	\(0.199147\pi\)
							−0.972992	+	0.230839i	\(0.925853\pi\)
\(29\)	−25.7707	+	38.5686i	−0.888644	+	1.32995i	0.0548271	+	0.998496i	\(0.482539\pi\)
							−0.943471	+	0.331454i	\(0.892461\pi\)
\(31\)	−33.2137	+	6.60662i	−1.07141	+	0.213117i	−0.699124	−	0.715001i	\(-0.746426\pi\)
							−0.372287	+	0.928118i	\(0.621426\pi\)
\(37\)	−8.93006	−	44.8944i	−0.241353	−	1.21336i	−0.891311	−	0.453393i	\(-0.850213\pi\)
							0.649958	−	0.759970i	\(-0.274787\pi\)
\(41\)	7.15212	−	4.77889i	0.174442	−	0.116558i	−0.465284	−	0.885161i	\(-0.654048\pi\)
							0.639726	+	0.768603i	\(0.279048\pi\)
\(43\)	−32.2691	+	13.3663i	−0.750443	+	0.310844i	−0.724923	−	0.688830i	\(-0.758124\pi\)
							−0.0255210	+	0.999674i	\(0.508124\pi\)
\(47\)	−55.2037	+	55.2037i	−1.17455	+	1.17455i	−0.193434	+	0.981113i	\(0.561962\pi\)
							−0.981113	+	0.193434i	\(0.938038\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	170.3.p.a.11.1	✓	48
17.14	odd	16	inner	170.3.p.a.31.1	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.3.p.a.11.1	✓	48	1.1	even	1	trivial
170.3.p.a.31.1	yes	48	17.14	odd	16	inner