Embedded newform 287.2.w.a.3.19

• Label: 287.2.w.a.3.19
• 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			3.19
Character	\(\chi\)	\(=\)	287.3
Dual form			287.2.w.a.96.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28562 + 0.344480i) q^{2} +(-1.94647 + 1.49358i) q^{3} +(-0.197909 - 0.114263i) q^{4} +(-2.20971 - 0.592091i) q^{5} +(-3.01691 + 1.24965i) q^{6} +(1.03924 - 2.43310i) q^{7} +(-2.09735 - 2.09735i) q^{8} +(0.781503 - 2.91661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 4 q^{2} - 12 q^{3} - 12 q^{5} - 8 q^{7} - 32 q^{8} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28562	+	0.344480i	0.909068	+	0.243584i	0.682906	−	0.730506i	\(-0.260716\pi\)
							0.226161	+	0.974090i	\(0.427382\pi\)
\(3\)	−1.94647	+	1.49358i	−1.12379	+	0.862316i	−0.991656	−	0.128915i	\(-0.958850\pi\)
							−0.132137	+	0.991231i	\(0.542184\pi\)
\(5\)	−2.20971	−	0.592091i	−0.988214	−	0.264791i	−0.271714	−	0.962378i	\(-0.587591\pi\)
							−0.716500	+	0.697587i	\(0.754257\pi\)
\(7\)	1.03924	−	2.43310i	0.392794	−	0.919626i
							\(11\)	−0.771244	+	0.591796i	−0.232539	+	0.178433i	−0.718458	−	0.695571i	\(-0.755152\pi\)
0.485919	+	0.874004i	\(0.338485\pi\)
\(13\)	−2.62710	−	1.08818i	−0.728627	−	0.301807i	−0.0126396	−	0.999920i	\(-0.504023\pi\)
							−0.715988	+	0.698113i	\(0.754023\pi\)
\(17\)	−0.418443	+	3.17839i	−0.101487	+	0.770873i	0.862712	+	0.505695i	\(0.168764\pi\)
							−0.964200	+	0.265178i	\(0.914569\pi\)
\(19\)	−4.21709	−	3.23589i	−0.967468	−	0.742364i	−0.00122550	−	0.999999i	\(-0.500390\pi\)
							−0.966242	+	0.257635i	\(0.917057\pi\)
\(23\)	2.28878	−	1.32143i	0.477243	−	0.275537i	−0.242024	−	0.970270i	\(-0.577811\pi\)
							0.719267	+	0.694734i	\(0.244478\pi\)
\(29\)	0.818609	−	1.97630i	0.152012	−	0.366989i	−0.829468	−	0.558554i	\(-0.811356\pi\)
							0.981480	+	0.191565i	\(0.0613563\pi\)
\(31\)	−5.41528	+	9.37955i	−0.972614	+	1.68462i	−0.285018	+	0.958522i	\(0.592000\pi\)
							−0.687596	+	0.726094i	\(0.741334\pi\)
\(37\)	−0.769681	−	1.33313i	−0.126535	−	0.219165i	0.795797	−	0.605563i	\(-0.207052\pi\)
							−0.922332	+	0.386399i	\(0.873719\pi\)
\(41\)	5.80727	+	2.69733i	0.906944	+	0.421252i
							\(43\)	−5.70919	+	5.70919i	−0.870643	+	0.870643i	−0.992542	−	0.121900i	\(-0.961101\pi\)
0.121900	+	0.992542i	\(0.461101\pi\)
\(47\)	0.197817	+	0.151790i	0.0288546	+	0.0221409i	0.623088	−	0.782152i	\(-0.285878\pi\)
							−0.594233	+	0.804293i	\(0.702544\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.w.a.3.19	✓	208
7.5	odd	6	inner	287.2.w.a.208.8	yes	208
41.14	odd	8	inner	287.2.w.a.178.8	yes	208
287.96	even	24	inner	287.2.w.a.96.19	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.w.a.3.19	✓	208	1.1	even	1	trivial
287.2.w.a.96.19	yes	208	287.96	even	24	inner
287.2.w.a.178.8	yes	208	41.14	odd	8	inner
287.2.w.a.208.8	yes	208	7.5	odd	6	inner