Embedded newform 287.2.z.a.4.8

• Label: 287.2.z.a.4.8
• Level: $287$
• Weight: $2$
• Character: 287.4
• 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.z (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(208\)
Relative dimension:	\(26\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.8
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41117 - 0.628293i) q^{2} +(0.158984 + 0.0917894i) q^{3} +(0.258386 + 0.286967i) q^{4} +(-1.25090 + 0.265887i) q^{5} +(-0.166683 - 0.229419i) q^{6} +(2.06082 + 1.65923i) q^{7} +(0.770360 + 2.37093i) q^{8} +(-1.48315 - 2.56889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41117	−	0.628293i	−0.997848	−	0.444270i	−0.158203	−	0.987407i	\(-0.550570\pi\)
							−0.839645	+	0.543136i	\(0.817237\pi\)
\(3\)	0.158984	+	0.0917894i	0.0917894	+	0.0529946i	0.545192	−	0.838311i	\(-0.316457\pi\)
							−0.453403	+	0.891306i	\(0.649790\pi\)
\(5\)	−1.25090	+	0.265887i	−0.559419	+	0.118908i	−0.478939	−	0.877848i	\(-0.658979\pi\)
							−0.0804800	+	0.996756i	\(0.525645\pi\)
\(7\)	2.06082	+	1.65923i	0.778915	+	0.627130i
							\(11\)	−0.0702662	+	0.330576i	−0.0211861	+	0.0996725i	−0.987470	−	0.157805i	\(-0.949558\pi\)
0.966284	+	0.257478i	\(0.0828915\pi\)
\(13\)	3.85547	+	5.30661i	1.06932	+	1.47179i	0.870767	+	0.491697i	\(0.163623\pi\)
							0.198550	+	0.980091i	\(0.436377\pi\)
\(17\)	1.09171	−	5.13611i	0.264779	−	1.24569i	−0.621823	−	0.783158i	\(-0.713608\pi\)
							0.886603	−	0.462532i	\(-0.153059\pi\)
\(19\)	4.42500	+	0.465087i	1.01517	+	0.106698i	0.597472	−	0.801890i	\(-0.296172\pi\)
							0.417694	+	0.908588i	\(0.362839\pi\)
\(23\)	4.77850	+	2.12753i	0.996387	+	0.443620i	0.839126	−	0.543937i	\(-0.183067\pi\)
							0.157261	+	0.987557i	\(0.449734\pi\)
\(29\)	3.57515	+	1.16164i	0.663889	+	0.215711i	0.621528	−	0.783392i	\(-0.286512\pi\)
							0.0423606	+	0.999102i	\(0.486512\pi\)
\(31\)	3.82398	+	0.812812i	0.686807	+	0.145985i	0.538081	−	0.842893i	\(-0.319149\pi\)
							0.148726	+	0.988878i	\(0.452483\pi\)
\(37\)	6.05437	−	1.28690i	0.995332	−	0.211564i	0.318676	−	0.947864i	\(-0.396762\pi\)
							0.676656	+	0.736299i	\(0.263429\pi\)
\(41\)	1.11240	−	6.30576i	0.173728	−	0.984794i
							\(43\)	2.68657	−	1.95191i	0.409698	−	0.297663i	−0.363782	−	0.931484i	\(-0.618515\pi\)
0.773480	+	0.633821i	\(0.218515\pi\)
\(47\)	−3.72416	+	8.36459i	−0.543224	+	1.22010i	0.408391	+	0.912807i	\(0.366090\pi\)
							−0.951615	+	0.307294i	\(0.900577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.8	✓	208
7.2	even	3	inner	287.2.z.a.86.19	yes	208
41.31	even	10	inner	287.2.z.a.277.19	yes	208
287.72	even	30	inner	287.2.z.a.72.8	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.8	✓	208	1.1	even	1	trivial
287.2.z.a.72.8	yes	208	287.72	even	30	inner
287.2.z.a.86.19	yes	208	7.2	even	3	inner
287.2.z.a.277.19	yes	208	41.31	even	10	inner