Embedded newform 287.2.r.c.9.19

• Label: 287.2.r.c.9.19
• 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			9.19
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.58851 + 0.917128i) q^{2} +(-0.0945920 + 0.353022i) q^{3} +(0.682248 + 1.18169i) q^{4} +(-0.406421 - 0.234647i) q^{5} +(-0.474027 + 0.474027i) q^{6} +(2.16453 + 1.52145i) q^{7} -1.16568i q^{8} +(2.48240 + 1.43321i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.58851	+	0.917128i	1.12325	+	0.648508i	0.942228	−	0.334972i	\(-0.108727\pi\)
							0.181020	+	0.983479i	\(0.442060\pi\)
\(3\)	−0.0945920	+	0.353022i	−0.0546127	+	0.203817i	−0.987841	−	0.155466i	\(-0.950312\pi\)
							0.933229	+	0.359283i	\(0.116979\pi\)
\(5\)	−0.406421	−	0.234647i	−0.181757	−	0.104937i	0.406361	−	0.913713i	\(-0.366798\pi\)
							−0.588118	+	0.808775i	\(0.700131\pi\)
\(7\)	2.16453	+	1.52145i	0.818114	+	0.575056i
							\(11\)	−0.963854	+	3.59715i	−0.290613	+	1.08458i	0.654026	+	0.756472i	\(0.273079\pi\)
−0.944639	+	0.328111i	\(0.893588\pi\)
\(13\)	0.00301715	+	0.00301715i	0.000836808	+	0.000836808i	0.707525	−	0.706688i	\(-0.249812\pi\)
							−0.706688	+	0.707525i	\(0.749812\pi\)
\(17\)	−4.12547	−	1.10542i	−1.00057	−	0.268103i	−0.278888	−	0.960324i	\(-0.589966\pi\)
							−0.721686	+	0.692221i	\(0.756632\pi\)
\(19\)	−0.562096	−	2.09777i	−0.128954	−	0.481262i	0.870996	−	0.491290i	\(-0.163475\pi\)
							−0.999950	+	0.0100282i	\(0.996808\pi\)
\(23\)	0.512968	−	0.888487i	0.106961	−	0.185262i	−0.807577	−	0.589763i	\(-0.799221\pi\)
							0.914538	+	0.404500i	\(0.132555\pi\)
\(29\)	−0.181387	−	0.181387i	−0.0336827	−	0.0336827i	0.690065	−	0.723748i	\(-0.257582\pi\)
							−0.723748	+	0.690065i	\(0.757582\pi\)
\(31\)	−2.89406	−	5.01266i	−0.519788	−	0.900300i	−0.999735	−	0.0230024i	\(-0.992677\pi\)
							0.479947	−	0.877297i	\(-0.340656\pi\)
\(37\)	2.24415	−	3.88698i	0.368936	−	0.639016i	−0.620463	−	0.784235i	\(-0.713055\pi\)
							0.989399	+	0.145219i	\(0.0463888\pi\)
\(41\)	−1.49534	−	6.22607i	−0.233533	−	0.972349i
							\(43\)			3.18338i			0.485461i	0.970094	+	0.242731i	\(0.0780431\pi\)
−0.970094	+	0.242731i	\(0.921957\pi\)
\(47\)	−0.228148	−	0.851459i	−0.0332788	−	0.124198i	0.947288	−	0.320383i	\(-0.103811\pi\)
							−0.980567	+	0.196185i	\(0.937145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.19	✓	96
7.4	even	3	inner	287.2.r.c.214.6	yes	96
41.32	even	4	inner	287.2.r.c.114.6	yes	96
287.32	even	12	inner	287.2.r.c.32.19	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.19	✓	96	1.1	even	1	trivial
287.2.r.c.32.19	yes	96	287.32	even	12	inner
287.2.r.c.114.6	yes	96	41.32	even	4	inner
287.2.r.c.214.6	yes	96	7.4	even	3	inner