Embedded newform 287.2.w.a.3.3

• Label: 287.2.w.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.32636 - 0.623346i) q^{2} +(-0.340938 + 0.261611i) q^{3} +(3.29134 + 1.90025i) q^{4} +(4.19521 + 1.12410i) q^{5} +(0.956220 - 0.396079i) q^{6} +(2.58749 - 0.552173i) q^{7} +(-3.06629 - 3.06629i) q^{8} +(-0.728659 + 2.71939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.32636	−	0.623346i	−1.64498	−	0.440772i	−0.686782	−	0.726863i	\(-0.740977\pi\)
							−0.958202	+	0.286091i	\(0.907644\pi\)
\(3\)	−0.340938	+	0.261611i	−0.196841	+	0.151041i	−0.702479	−	0.711704i	\(-0.747924\pi\)
							0.505638	+	0.862746i	\(0.331257\pi\)
\(5\)	4.19521	+	1.12410i	1.87615	+	0.502714i	0.999778	+	0.0210851i	\(0.00671210\pi\)
							0.876375	+	0.481629i	\(0.159955\pi\)
\(7\)	2.58749	−	0.552173i	0.977979	−	0.208702i
							\(11\)	−1.67598	+	1.28603i	−0.505328	+	0.387752i	−0.829728	−	0.558168i	\(-0.811504\pi\)
0.324400	+	0.945920i	\(0.394838\pi\)
\(13\)	0.575230	+	0.238268i	0.159540	+	0.0660836i	0.461024	−	0.887387i	\(-0.347482\pi\)
							−0.301484	+	0.953471i	\(0.597482\pi\)
\(17\)	−0.728907	+	5.53660i	−0.176786	+	1.34282i	0.641107	+	0.767452i	\(0.278476\pi\)
							−0.817893	+	0.575371i	\(0.804858\pi\)
\(19\)	−5.21505	−	4.00165i	−1.19641	−	0.918041i	−0.198344	−	0.980132i	\(-0.563556\pi\)
							−0.998070	+	0.0620910i	\(0.980223\pi\)
\(23\)	1.41575	−	0.817381i	0.295203	−	0.170436i	−0.345083	−	0.938572i	\(-0.612149\pi\)
							0.640286	+	0.768137i	\(0.278816\pi\)
\(29\)	0.748636	−	1.80737i	0.139018	−	0.335620i	−0.839002	−	0.544128i	\(-0.816861\pi\)
							0.978021	+	0.208508i	\(0.0668607\pi\)
\(31\)	−0.453063	+	0.784728i	−0.0813725	+	0.140941i	−0.903840	−	0.427871i	\(-0.859264\pi\)
							0.822467	+	0.568813i	\(0.192597\pi\)
\(37\)	−2.67381	−	4.63118i	−0.439572	−	0.761361i	0.558084	−	0.829784i	\(-0.311537\pi\)
							−0.997656	+	0.0684229i	\(0.978203\pi\)
\(41\)	−6.39112	−	0.391875i	−0.998125	−	0.0612006i
							\(43\)	4.29414	−	4.29414i	0.654850	−	0.654850i	−0.299307	−	0.954157i	\(-0.596755\pi\)
0.954157	+	0.299307i	\(0.0967554\pi\)
\(47\)	6.31814	+	4.84808i	0.921595	+	0.707165i	0.956348	−	0.292231i	\(-0.0943977\pi\)
							−0.0347529	+	0.999396i	\(0.511064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.3	✓	208
7.5	odd	6	inner	287.2.w.a.208.24	yes	208
41.14	odd	8	inner	287.2.w.a.178.24	yes	208
287.96	even	24	inner	287.2.w.a.96.3	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.3	✓	208	1.1	even	1	trivial
287.2.w.a.96.3	yes	208	287.96	even	24	inner
287.2.w.a.178.24	yes	208	41.14	odd	8	inner
287.2.w.a.208.24	yes	208	7.5	odd	6	inner