Embedded newform 287.2.w.a.3.12

• Label: 287.2.w.a.3.12
• 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			3.12
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.389415 - 0.104344i) q^{2} +(-0.138074 + 0.105948i) q^{3} +(-1.59129 - 0.918734i) q^{4} +(-2.37555 - 0.636527i) q^{5} +(0.0648230 - 0.0268506i) q^{6} +(2.38380 + 1.14782i) q^{7} +(1.09395 + 1.09395i) q^{8} +(-0.768618 + 2.86852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.389415	−	0.104344i	−0.275358	−	0.0737820i	0.118497	−	0.992954i	\(-0.462192\pi\)
							−0.393856	+	0.919172i	\(0.628859\pi\)
\(3\)	−0.138074	+	0.105948i	−0.0797169	+	0.0611689i	−0.647851	−	0.761767i	\(-0.724332\pi\)
							0.568134	+	0.822936i	\(0.307665\pi\)
\(5\)	−2.37555	−	0.636527i	−1.06238	−	0.284663i	−0.315020	−	0.949085i	\(-0.602011\pi\)
							−0.747358	+	0.664422i	\(0.768678\pi\)
\(7\)	2.38380	+	1.14782i	0.900992	+	0.433836i
							\(11\)	−1.74711	+	1.34061i	−0.526774	+	0.404208i	−0.837621	−	0.546252i	\(-0.816054\pi\)
0.310847	+	0.950460i	\(0.399387\pi\)
\(13\)	3.55654	+	1.47317i	0.986406	+	0.408583i	0.816795	−	0.576928i	\(-0.195749\pi\)
							0.169611	+	0.985511i	\(0.445749\pi\)
\(17\)	−0.345975	+	2.62794i	−0.0839112	+	0.637369i	0.896875	+	0.442284i	\(0.145832\pi\)
							−0.980786	+	0.195085i	\(0.937502\pi\)
\(19\)	2.29435	+	1.76051i	0.526359	+	0.403890i	0.837469	−	0.546484i	\(-0.184034\pi\)
							−0.311110	+	0.950374i	\(0.600701\pi\)
\(23\)	−3.05315	+	1.76274i	−0.636625	+	0.367556i	−0.783313	−	0.621627i	\(-0.786472\pi\)
							0.146688	+	0.989183i	\(0.453139\pi\)
\(29\)	0.962381	−	2.32339i	0.178710	−	0.431443i	−0.808987	−	0.587827i	\(-0.799984\pi\)
							0.987696	+	0.156384i	\(0.0499836\pi\)
\(31\)	−4.90072	+	8.48829i	−0.880195	+	1.52454i	−0.0290706	+	0.999577i	\(0.509255\pi\)
							−0.851124	+	0.524965i	\(0.824079\pi\)
\(37\)	−3.61530	−	6.26188i	−0.594351	−	1.02945i	−0.993638	−	0.112620i	\(-0.964076\pi\)
							0.399287	−	0.916826i	\(-0.369258\pi\)
\(41\)	−0.293306	+	6.39640i	−0.0458068	+	0.998950i
							\(43\)	1.30809	−	1.30809i	0.199482	−	0.199482i	−0.600296	−	0.799778i	\(-0.704950\pi\)
0.799778	+	0.600296i	\(0.204950\pi\)
\(47\)	0.694784	+	0.533126i	0.101345	+	0.0777644i	0.658191	−	0.752851i	\(-0.271322\pi\)
							−0.556846	+	0.830616i	\(0.687989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.12	✓	208
7.5	odd	6	inner	287.2.w.a.208.15	yes	208
41.14	odd	8	inner	287.2.w.a.178.15	yes	208
287.96	even	24	inner	287.2.w.a.96.12	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.12	✓	208	1.1	even	1	trivial
287.2.w.a.96.12	yes	208	287.96	even	24	inner
287.2.w.a.178.15	yes	208	41.14	odd	8	inner
287.2.w.a.208.15	yes	208	7.5	odd	6	inner