Embedded newform 287.3.y.a.10.10

• Label: 287.3.y.a.10.10
• Level: $287$
• Weight: $3$
• Character: 287.10
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.y (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.10
Character	\(\chi\)	\(=\)	287.10
Dual form			287.3.y.a.201.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.282410 + 2.68695i) q^{2} +(-1.99951 - 1.15441i) q^{3} +(-3.22738 - 0.686000i) q^{4} +(0.0649077 + 0.0584432i) q^{5} +(3.66654 - 5.04656i) q^{6} +(-6.98405 + 0.472222i) q^{7} +(-0.584859 + 1.80001i) q^{8} +(-1.83465 - 3.17771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} - 24 q^{3} + 101 q^{4} - 9 q^{5} - 6 q^{7} - 16 q^{8} + 576 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.282410	+	2.68695i	−0.141205	+	1.34348i	0.662774	+	0.748819i	\(0.269379\pi\)
							−0.803979	+	0.594657i	\(0.797288\pi\)
\(3\)	−1.99951	−	1.15441i	−0.666502	−	0.384805i	0.128248	−	0.991742i	\(-0.459065\pi\)
							−0.794750	+	0.606937i	\(0.792398\pi\)
\(5\)	0.0649077	+	0.0584432i	0.0129815	+	0.0116886i	0.675596	−	0.737272i	\(-0.263886\pi\)
							−0.662614	+	0.748961i	\(0.730553\pi\)
\(7\)	−6.98405	+	0.472222i	−0.997722	+	0.0674603i
							\(11\)	4.71986	+	5.24193i	0.429078	+	0.476539i	0.918451	−	0.395535i	\(-0.129441\pi\)
−0.489373	+	0.872075i	\(0.662774\pi\)
\(13\)	3.53828	−	4.87003i	0.272176	−	0.374618i	−0.650947	−	0.759123i	\(-0.725628\pi\)
							0.923123	+	0.384505i	\(0.125628\pi\)
\(17\)	9.67762	−	8.71377i	0.569272	−	0.512575i	−0.333468	−	0.942761i	\(-0.608219\pi\)
							0.902740	+	0.430187i	\(0.141552\pi\)
\(19\)	11.1354	−	25.0106i	0.586075	−	1.31635i	−0.340503	−	0.940243i	\(-0.610597\pi\)
							0.926578	−	0.376102i	\(-0.122736\pi\)
\(23\)	3.27036	−	31.1154i	0.142189	−	1.35284i	−0.657966	−	0.753048i	\(-0.728583\pi\)
							0.800155	−	0.599793i	\(-0.204751\pi\)
\(29\)	10.6039	+	32.6355i	0.365652	+	1.12536i	0.949572	+	0.313549i	\(0.101518\pi\)
							−0.583920	+	0.811811i	\(0.698482\pi\)
\(31\)	7.34437	−	6.61290i	0.236915	−	0.213319i	−0.542118	−	0.840303i	\(-0.682377\pi\)
							0.779033	+	0.626983i	\(0.215711\pi\)
\(37\)	−10.1227	+	11.2424i	−0.273586	+	0.303848i	−0.864243	−	0.503075i	\(-0.832202\pi\)
							0.590657	+	0.806923i	\(0.298869\pi\)
\(41\)	3.40809	−	40.8581i	0.0831242	−	0.996539i
							\(43\)	−29.0708	−	21.1212i	−0.676065	−	0.491190i	0.195985	−	0.980607i	\(-0.437210\pi\)
−0.872050	+	0.489417i	\(0.837210\pi\)
\(47\)	−28.0065	−	2.94360i	−0.595883	−	0.0626299i	−0.198214	−	0.980159i	\(-0.563514\pi\)
							−0.397669	+	0.917529i	\(0.630181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.y.a.10.10	✓	432
7.5	odd	6	inner	287.3.y.a.215.45	yes	432
41.37	even	5	inner	287.3.y.a.283.45	yes	432
287.201	odd	30	inner	287.3.y.a.201.10	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.y.a.10.10	✓	432	1.1	even	1	trivial
287.3.y.a.201.10	yes	432	287.201	odd	30	inner
287.3.y.a.215.45	yes	432	7.5	odd	6	inner
287.3.y.a.283.45	yes	432	41.37	even	5	inner