Embedded newform 287.2.r.c.9.3

• Label: 287.2.r.c.9.3
• Level: $287$
• Weight: $2$
• Character: 287.9
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.r (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			9.3
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94459 - 1.12271i) q^{2} +(-0.198974 + 0.742581i) q^{3} +(1.52096 + 2.63438i) q^{4} +(-1.21146 - 0.699435i) q^{5} +(1.22063 - 1.22063i) q^{6} +(-0.635224 - 2.56836i) q^{7} -2.33955i q^{8} +(2.08624 + 1.20449i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94459	−	1.12271i	−1.37503	−	0.793877i	−0.383478	−	0.923550i	\(-0.625274\pi\)
							−0.991557	+	0.129673i	\(0.958607\pi\)
\(3\)	−0.198974	+	0.742581i	−0.114878	+	0.428729i	−0.999278	−	0.0380003i	\(-0.987901\pi\)
							0.884400	+	0.466730i	\(0.154568\pi\)
\(5\)	−1.21146	−	0.699435i	−0.541780	−	0.312797i	0.204020	−	0.978967i	\(-0.434599\pi\)
							−0.745800	+	0.666170i	\(0.767933\pi\)
\(7\)	−0.635224	−	2.56836i	−0.240092	−	0.970750i
							\(11\)	−0.212677	+	0.793721i	−0.0641245	+	0.239316i	−0.990548	−	0.137167i	\(-0.956200\pi\)
0.926423	+	0.376483i	\(0.122867\pi\)
\(13\)	−0.565290	−	0.565290i	−0.156783	−	0.156783i	0.624356	−	0.781140i	\(-0.285361\pi\)
							−0.781140	+	0.624356i	\(0.785361\pi\)
\(17\)	−7.02243	−	1.88165i	−1.70319	−	0.456368i	−0.729450	−	0.684034i	\(-0.760224\pi\)
							−0.973739	+	0.227666i	\(0.926891\pi\)
\(19\)	−1.66108	−	6.19924i	−0.381078	−	1.42220i	−0.844257	−	0.535938i	\(-0.819958\pi\)
							0.463179	−	0.886265i	\(-0.346709\pi\)
\(23\)	−4.40605	+	7.63150i	−0.918724	+	1.59128i	−0.117369	+	0.993088i	\(0.537446\pi\)
							−0.801355	+	0.598189i	\(0.795887\pi\)
\(29\)	−3.47304	−	3.47304i	−0.644928	−	0.644928i	0.306835	−	0.951763i	\(-0.400730\pi\)
							−0.951763	+	0.306835i	\(0.900730\pi\)
\(31\)	0.0346274	+	0.0599765i	0.00621927	+	0.0107721i	0.869118	−	0.494604i	\(-0.164687\pi\)
							−0.862899	+	0.505376i	\(0.831354\pi\)
\(37\)	−3.81546	+	6.60857i	−0.627258	+	1.08644i	0.360842	+	0.932627i	\(0.382489\pi\)
							−0.988100	+	0.153815i	\(0.950844\pi\)
\(41\)	1.44288	−	6.23844i	0.225339	−	0.974280i
							\(43\)			0.891443i			0.135944i	0.997687	+	0.0679719i	\(0.0216528\pi\)
−0.997687	+	0.0679719i	\(0.978347\pi\)
\(47\)	−1.38807	−	5.18034i	−0.202471	−	0.755630i	−0.990206	−	0.139616i	\(-0.955413\pi\)
							0.787735	−	0.616014i	\(-0.211254\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.3	✓	96
7.4	even	3	inner	287.2.r.c.214.22	yes	96
41.32	even	4	inner	287.2.r.c.114.22	yes	96
287.32	even	12	inner	287.2.r.c.32.3	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.3	✓	96	1.1	even	1	trivial
287.2.r.c.32.3	yes	96	287.32	even	12	inner
287.2.r.c.114.22	yes	96	41.32	even	4	inner
287.2.r.c.214.22	yes	96	7.4	even	3	inner