Embedded newform 176.3.k.a.67.16

• Label: 176.3.k.a.67.16
• Level: $176$
• Weight: $3$
• Character: 176.67
• Analytic conductor: $4.796$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	176.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.79565265274\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			67.16
Character	\(\chi\)	\(=\)	176.67
Dual form			176.3.k.a.155.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.670141 + 1.88439i) q^{2} +(1.36978 - 1.36978i) q^{3} +(-3.10182 - 2.52561i) q^{4} +(-2.05346 + 2.05346i) q^{5} +(1.66325 + 3.49914i) q^{6} +6.31785 q^{7} +(6.83788 - 4.15252i) q^{8} +5.24741i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 12 q^{6} - 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\)	\(111\)	\(133\)	\(145\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.670141	+	1.88439i	−0.335070	+	0.942193i
							\(3\)	1.36978	−	1.36978i	0.456593	−	0.456593i	−0.440942	−	0.897535i	\(-0.645356\pi\)
0.897535	+	0.440942i	\(0.145356\pi\)
\(5\)	−2.05346	+	2.05346i	−0.410691	+	0.410691i	−0.881979	−	0.471288i	\(-0.843789\pi\)
							0.471288	+	0.881979i	\(0.343789\pi\)
\(7\)	6.31785			0.902551			0.451275	−	0.892385i	\(-0.350969\pi\)
							0.451275	+	0.892385i	\(0.350969\pi\)
\(11\)	−2.34521	−	2.34521i	−0.213201	−	0.213201i
							\(13\)	17.4408	+	17.4408i	1.34160	+	1.34160i	0.894469	+	0.447131i	\(0.147554\pi\)
0.447131	+	0.894469i	\(0.352446\pi\)
\(17\)	8.61576			0.506809			0.253405	−	0.967360i	\(-0.418450\pi\)
							0.253405	+	0.967360i	\(0.418450\pi\)
\(19\)	−6.41688	+	6.41688i	−0.337730	+	0.337730i	−0.855513	−	0.517782i	\(-0.826758\pi\)
							0.517782	+	0.855513i	\(0.326758\pi\)
\(23\)	26.2643			1.14193			0.570964	−	0.820975i	\(-0.306570\pi\)
							0.570964	+	0.820975i	\(0.306570\pi\)
\(29\)	−40.2020	−	40.2020i	−1.38628	−	1.38628i	−0.832995	−	0.553280i	\(-0.813376\pi\)
							−0.553280	−	0.832995i	\(-0.686624\pi\)
\(31\)		−	16.0467i		−	0.517635i	−0.965926	−	0.258818i	\(-0.916667\pi\)
							0.965926	−	0.258818i	\(-0.0833329\pi\)
\(37\)	−27.3949	+	27.3949i	−0.740401	+	0.740401i	−0.972655	−	0.232254i	\(-0.925390\pi\)
							0.232254	+	0.972655i	\(0.425390\pi\)
\(41\)		−	20.5331i		−	0.500808i	−0.968141	−	0.250404i	\(-0.919437\pi\)
							0.968141	−	0.250404i	\(-0.0805634\pi\)
\(43\)	44.5477	+	44.5477i	1.03599	+	1.03599i	0.999328	+	0.0366653i	\(0.0116736\pi\)
							0.0366653	+	0.999328i	\(0.488326\pi\)
\(47\)		−	45.5203i		−	0.968516i	−0.874925	−	0.484258i	\(-0.839090\pi\)
							0.874925	−	0.484258i	\(-0.160910\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.3.k.a.67.16	✓	80
4.3	odd	2		704.3.k.a.111.14		80
16.5	even	4		704.3.k.a.463.14		80
16.11	odd	4	inner	176.3.k.a.155.16	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.3.k.a.67.16	✓	80	1.1	even	1	trivial
176.3.k.a.155.16	yes	80	16.11	odd	4	inner
704.3.k.a.111.14		80	4.3	odd	2
704.3.k.a.463.14		80	16.5	even	4