Embedded newform 287.2.w.a.3.2

• Label: 287.2.w.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46478 - 0.660435i) q^{2} +(2.16154 - 1.65861i) q^{3} +(3.90691 + 2.25565i) q^{4} +(1.76092 + 0.471838i) q^{5} +(-6.42311 + 2.66054i) q^{6} +(-2.53259 - 0.765512i) q^{7} +(-4.53126 - 4.53126i) q^{8} +(1.14481 - 4.27250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46478	−	0.660435i	−1.74286	−	0.466998i	−0.759782	−	0.650178i	\(-0.774694\pi\)
							−0.983079	+	0.183180i	\(0.941361\pi\)
\(3\)	2.16154	−	1.65861i	1.24796	−	0.957597i	0.248027	−	0.968753i	\(-0.420218\pi\)
							0.999938	+	0.0111565i	\(0.00355129\pi\)
\(5\)	1.76092	+	0.471838i	0.787509	+	0.211012i	0.630092	−	0.776520i	\(-0.283017\pi\)
							0.157416	+	0.987532i	\(0.449684\pi\)
\(7\)	−2.53259	−	0.765512i	−0.957228	−	0.289336i
							\(11\)	3.02210	−	2.31894i	0.911197	−	0.699186i	−0.0427737	−	0.999085i	\(-0.513619\pi\)
0.953971	+	0.299898i	\(0.0969528\pi\)
\(13\)	−2.90927	−	1.20506i	−0.806886	−	0.334223i	−0.0591752	−	0.998248i	\(-0.518847\pi\)
							−0.747711	+	0.664024i	\(0.768847\pi\)
\(17\)	−0.177126	+	1.34540i	−0.0429593	+	0.326308i	0.956449	+	0.291901i	\(0.0942877\pi\)
							−0.999408	+	0.0344074i	\(0.989046\pi\)
\(19\)	2.53187	+	1.94278i	0.580852	+	0.445703i	0.856931	−	0.515432i	\(-0.172368\pi\)
							−0.276079	+	0.961135i	\(0.589035\pi\)
\(23\)	6.70827	−	3.87302i	1.39877	−	0.807581i	0.404508	−	0.914535i	\(-0.367443\pi\)
							0.994264	+	0.106953i	\(0.0341096\pi\)
\(29\)	3.79559	−	9.16338i	0.704824	−	1.70160i	−0.00772141	−	0.999970i	\(-0.502458\pi\)
							0.712546	−	0.701626i	\(-0.247542\pi\)
\(31\)	−2.20764	+	3.82374i	−0.396503	+	0.686764i	−0.993292	−	0.115635i	\(-0.963110\pi\)
							0.596788	+	0.802399i	\(0.296443\pi\)
\(37\)	0.313860	+	0.543621i	0.0515982	+	0.0893707i	0.890671	−	0.454649i	\(-0.150235\pi\)
							−0.839073	+	0.544019i	\(0.816902\pi\)
\(41\)	−1.04955	+	6.31652i	−0.163912	+	0.986475i
							\(43\)	−6.72299	+	6.72299i	−1.02525	+	1.02525i	−0.0255727	+	0.999673i	\(0.508141\pi\)
−0.999673	+	0.0255727i	\(0.991859\pi\)
\(47\)	−7.19727	−	5.52266i	−1.04983	−	0.805563i	−0.0684529	−	0.997654i	\(-0.521806\pi\)
							−0.981377	+	0.192091i	\(0.938473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.2	✓	208
7.5	odd	6	inner	287.2.w.a.208.25	yes	208
41.14	odd	8	inner	287.2.w.a.178.25	yes	208
287.96	even	24	inner	287.2.w.a.96.2	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.2	✓	208	1.1	even	1	trivial
287.2.w.a.96.2	yes	208	287.96	even	24	inner
287.2.w.a.178.25	yes	208	41.14	odd	8	inner
287.2.w.a.208.25	yes	208	7.5	odd	6	inner