Embedded newform 287.3.ba.a.15.9

• Label: 287.3.ba.a.15.9
• Level: $287$
• Weight: $3$
• Character: 287.15
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(42\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.9
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.70944 - 0.429133i) q^{2} +(-3.73900 + 1.54874i) q^{3} +(3.35268 + 1.08935i) q^{4} +(2.16823 + 4.25539i) q^{5} +(10.7952 - 2.59170i) q^{6} +(-2.57265 - 0.617638i) q^{7} +(1.16048 + 0.591295i) q^{8} +(5.21755 - 5.21755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.70944	−	0.429133i	−1.35472	−	0.214566i	−0.563509	−	0.826110i	\(-0.690549\pi\)
							−0.791211	+	0.611544i	\(0.790549\pi\)
\(3\)	−3.73900	+	1.54874i	−1.24633	+	0.516248i	−0.905688	−	0.423944i	\(-0.860645\pi\)
							−0.340645	+	0.940192i	\(0.610645\pi\)
\(5\)	2.16823	+	4.25539i	0.433646	+	0.851079i	0.999644	+	0.0266722i	\(0.00849103\pi\)
							−0.565998	+	0.824407i	\(0.691509\pi\)
\(7\)	−2.57265	−	0.617638i	−0.367521	−	0.0882341i
							\(11\)	8.50648	−	7.26522i	0.773317	−	0.660475i	−0.172561	−	0.984999i	\(-0.555204\pi\)
0.945877	+	0.324524i	\(0.105204\pi\)
\(13\)	1.37638	−	2.24604i	0.105875	−	0.172773i	−0.795315	−	0.606196i	\(-0.792695\pi\)
							0.901190	+	0.433424i	\(0.142695\pi\)
\(17\)	1.59422	−	20.2565i	0.0937777	−	1.19156i	−0.752418	−	0.658686i	\(-0.771112\pi\)
							0.846195	−	0.532873i	\(-0.178888\pi\)
\(19\)	15.9019	+	25.9496i	0.836944	+	1.36577i	0.929025	+	0.370017i	\(0.120648\pi\)
							−0.0920805	+	0.995752i	\(0.529352\pi\)
\(23\)	−12.2937	−	16.9208i	−0.534509	−	0.735689i	0.453300	−	0.891358i	\(-0.350247\pi\)
							−0.987809	+	0.155669i	\(0.950247\pi\)
\(29\)	4.42112	+	56.1756i	0.152452	+	1.93709i	0.312370	+	0.949960i	\(0.398877\pi\)
							−0.159918	+	0.987130i	\(0.551123\pi\)
\(31\)	51.0019	−	16.5715i	1.64522	−	0.534565i	0.667526	−	0.744587i	\(-0.267353\pi\)
							0.977697	+	0.210022i	\(0.0673534\pi\)
\(37\)	−19.5192	+	60.0739i	−0.527546	+	1.62362i	0.231681	+	0.972792i	\(0.425578\pi\)
							−0.759226	+	0.650827i	\(0.774422\pi\)
\(41\)	−26.8363	+	30.9970i	−0.654543	+	0.756025i
							\(43\)	31.4318	+	4.97830i	0.730971	+	0.115774i	0.510816	−	0.859690i	\(-0.329344\pi\)
0.220156	+	0.975465i	\(0.429344\pi\)
\(47\)	15.3895	−	3.69470i	0.327437	−	0.0786106i	−0.0663943	−	0.997793i	\(-0.521150\pi\)
							0.393831	+	0.919183i	\(0.371150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.9	✓	672
41.11	odd	40	inner	287.3.ba.a.134.9	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.9	✓	672	1.1	even	1	trivial
287.3.ba.a.134.9	yes	672	41.11	odd	40	inner