Embedded newform 833.2.bl.a.2.9

• Label: 833.2.bl.a.2.9
• Level: $833$
• Weight: $2$
• Character: 833.2
• Analytic conductor: $6.652$
• Analytic rank: $0$
• Dimension: $3936$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 833 = 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	833.bl (of order \(168\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.65153848837\)
Analytic rank:	\(0\)
Dimension:	\(3936\)
Relative dimension:	\(82\) over \(\Q(\zeta_{168})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{168}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	833.2
Dual form			833.2.bl.a.417.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07775 - 2.03921i) q^{2} +(0.0334431 + 1.78820i) q^{3} +(-1.87018 + 2.74304i) q^{4} +(1.19245 + 1.97898i) q^{5} +(3.61046 - 1.99543i) q^{6} +(-0.295065 + 2.62925i) q^{7} +(3.02524 + 0.340862i) q^{8} +(-0.198621 + 0.00743189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3936 q - 52 q^{2} - 52 q^{3} - 56 q^{5} - 40 q^{6} - 48 q^{7} - 24 q^{8} - 52 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\)	\(52\)	\(785\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07775	−	2.03921i	−0.762086	−	1.44194i	−0.892770	−	0.450513i	\(-0.851241\pi\)
							0.130684	−	0.991424i	\(-0.458283\pi\)
\(3\)	0.0334431	+	1.78820i	0.0193084	+	1.03241i	0.865914	+	0.500194i	\(0.166738\pi\)
							−0.846605	+	0.532221i	\(0.821357\pi\)
\(5\)	1.19245	+	1.97898i	0.533281	+	0.885028i	1.00000		0.000621945i	\(0.000197971\pi\)
							−0.466719	+	0.884406i	\(0.654564\pi\)
\(7\)	−0.295065	+	2.62925i	−0.111524	+	0.993762i
							\(11\)	2.61977	−	0.972052i	0.789889	−	0.293085i	0.0794458	−	0.996839i	\(-0.474685\pi\)
0.710443	+	0.703754i	\(0.248494\pi\)
\(13\)	−0.441024	+	0.100661i	−0.122318	+	0.0279183i	−0.283242	−	0.959049i	\(-0.591410\pi\)
							0.160924	+	0.986967i	\(0.448553\pi\)
\(17\)	−0.478950	−	4.09519i	−0.116162	−	0.993230i
							\(19\)	5.84151	−	1.56523i	1.34013	−	0.359088i	0.483650	−	0.875261i	\(-0.339311\pi\)
0.856484	+	0.516173i	\(0.172644\pi\)
\(23\)	−3.76167	+	7.45145i	−0.784362	+	1.55373i	0.0475774	+	0.998868i	\(0.484850\pi\)
							−0.831939	+	0.554867i	\(0.812769\pi\)
\(29\)	0.275118	−	4.89893i	0.0510881	−	0.909708i	−0.863255	−	0.504768i	\(-0.831578\pi\)
							0.914343	−	0.404940i	\(-0.132708\pi\)
\(31\)	3.48624	+	2.67509i	0.626148	+	0.480460i	0.872431	−	0.488738i	\(-0.162543\pi\)
							−0.246283	+	0.969198i	\(0.579209\pi\)
\(37\)	−6.52108	+	9.95984i	−1.07206	+	1.63739i	−0.363732	+	0.931504i	\(0.618498\pi\)
							−0.708328	+	0.705884i	\(0.750550\pi\)
\(41\)	−7.47548	+	2.15365i	−1.16747	+	0.336344i	−0.803638	−	0.595119i	\(-0.797105\pi\)
							−0.363837	+	0.931463i	\(0.618533\pi\)
\(43\)	1.13563	+	10.0790i	0.173182	+	1.53703i	0.716547	+	0.697538i	\(0.245721\pi\)
							−0.543365	+	0.839496i	\(0.682850\pi\)
\(47\)	2.09385	−	6.78809i	0.305419	−	0.990145i	−0.664617	−	0.747184i	\(-0.731405\pi\)
							0.970036	−	0.242961i	\(-0.0781186\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	833.2.bl.a.2.9	✓	3936
17.9	even	8	inner	833.2.bl.a.247.9	yes	3936
49.25	even	21	inner	833.2.bl.a.172.9	yes	3936
833.417	even	168	inner	833.2.bl.a.417.9	yes	3936

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
833.2.bl.a.2.9	✓	3936	1.1	even	1	trivial
833.2.bl.a.172.9	yes	3936	49.25	even	21	inner
833.2.bl.a.247.9	yes	3936	17.9	even	8	inner
833.2.bl.a.417.9	yes	3936	833.417	even	168	inner