Embedded newform 287.2.w.a.3.1

• Label: 287.2.w.a.3.1
• Level: $287$
• Weight: $2$
• Character: 287.3
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $208$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.w (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.56100 - 0.686218i) q^{2} +(-0.904761 + 0.694247i) q^{3} +(4.35578 + 2.51481i) q^{4} +(-1.05700 - 0.283223i) q^{5} +(2.79350 - 1.15711i) q^{6} +(-0.803969 + 2.52064i) q^{7} +(-5.67989 - 5.67989i) q^{8} +(-0.439844 + 1.64152i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.56100	−	0.686218i	−1.81090	−	0.485230i	−0.815310	−	0.579025i	\(-0.803433\pi\)
							−0.995591	+	0.0937954i	\(0.970100\pi\)
\(3\)	−0.904761	+	0.694247i	−0.522364	+	0.400824i	−0.836008	−	0.548717i	\(-0.815117\pi\)
							0.313645	+	0.949540i	\(0.398450\pi\)
\(5\)	−1.05700	−	0.283223i	−0.472706	−	0.126661i	0.0145985	−	0.999893i	\(-0.495353\pi\)
							−0.487304	+	0.873232i	\(0.662020\pi\)
\(7\)	−0.803969	+	2.52064i	−0.303872	+	0.952713i
							\(11\)	2.19825	−	1.68677i	0.662796	−	0.508581i	−0.221708	−	0.975113i	\(-0.571163\pi\)
0.884504	+	0.466532i	\(0.154497\pi\)
\(13\)	0.542496	+	0.224709i	0.150461	+	0.0623231i	0.456643	−	0.889650i	\(-0.349052\pi\)
							−0.306182	+	0.951973i	\(0.599052\pi\)
\(17\)	0.528491	−	4.01429i	0.128178	−	0.973608i	−0.799615	−	0.600513i	\(-0.794963\pi\)
							0.927793	−	0.373095i	\(-0.121704\pi\)
\(19\)	−1.64810	−	1.26464i	−0.378101	−	0.290127i	0.402174	−	0.915563i	\(-0.368255\pi\)
							−0.780276	+	0.625436i	\(0.784921\pi\)
\(23\)	−6.65420	+	3.84180i	−1.38750	+	0.801071i	−0.993033	−	0.117841i	\(-0.962403\pi\)
							−0.394463	+	0.918912i	\(0.629069\pi\)
\(29\)	−2.25479	+	5.44353i	−0.418703	+	1.01084i	0.564021	+	0.825761i	\(0.309254\pi\)
							−0.982724	+	0.185078i	\(0.940746\pi\)
\(31\)	−2.49150	+	4.31541i	−0.447487	+	0.775070i	−0.998222	−	0.0596100i	\(-0.981014\pi\)
							0.550735	+	0.834680i	\(0.314348\pi\)
\(37\)	−2.66968	−	4.62402i	−0.438892	−	0.760183i	0.558712	−	0.829362i	\(-0.311296\pi\)
							−0.997604	+	0.0691782i	\(0.977962\pi\)
\(41\)	0.199652	−	6.40001i	0.0311805	−	0.999514i
							\(43\)	−1.43092	+	1.43092i	−0.218213	+	0.218213i	−0.807745	−	0.589532i	\(-0.799312\pi\)
0.589532	+	0.807745i	\(0.299312\pi\)
\(47\)	−9.45671	−	7.25639i	−1.37940	−	1.05845i	−0.990550	−	0.137153i	\(-0.956205\pi\)
							−0.388853	−	0.921300i	\(-0.627129\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.1	✓	208
7.5	odd	6	inner	287.2.w.a.208.26	yes	208
41.14	odd	8	inner	287.2.w.a.178.26	yes	208
287.96	even	24	inner	287.2.w.a.96.1	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.1	✓	208	1.1	even	1	trivial
287.2.w.a.96.1	yes	208	287.96	even	24	inner
287.2.w.a.178.26	yes	208	41.14	odd	8	inner
287.2.w.a.208.26	yes	208	7.5	odd	6	inner