Embedded newform 287.2.z.a.4.19

• Label: 287.2.z.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.05975 + 0.471833i) q^{2} +(-2.06446 - 1.19192i) q^{3} +(-0.437810 - 0.486237i) q^{4} +(2.17853 - 0.463060i) q^{5} +(-1.62543 - 2.23722i) q^{6} +(-2.30952 + 1.29079i) q^{7} +(-0.951495 - 2.92840i) q^{8} +(1.34133 + 2.32325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.05975	+	0.471833i	0.749359	+	0.333636i	0.745633	−	0.666357i	\(-0.232147\pi\)
							0.00372608	+	0.999993i	\(0.498814\pi\)
\(3\)	−2.06446	−	1.19192i	−1.19192	−	0.688153i	−0.233175	−	0.972435i	\(-0.574911\pi\)
							−0.958741	+	0.284282i	\(0.908245\pi\)
\(5\)	2.17853	−	0.463060i	0.974266	−	0.207087i	0.306839	−	0.951762i	\(-0.400729\pi\)
							0.667427	+	0.744675i	\(0.267395\pi\)
\(7\)	−2.30952	+	1.29079i	−0.872916	+	0.487871i
							\(11\)	0.272923	−	1.28400i	0.0822893	−	0.387141i	−0.917658	−	0.397370i	\(-0.869923\pi\)
0.999948	+	0.0102295i	\(0.00325620\pi\)
\(13\)	−3.06161	−	4.21394i	−0.849137	−	1.16874i	−0.984052	−	0.177879i	\(-0.943076\pi\)
							0.134916	−	0.990857i	\(-0.456924\pi\)
\(17\)	1.10814	−	5.21339i	0.268764	−	1.26443i	−0.611992	−	0.790864i	\(-0.709632\pi\)
							0.880756	−	0.473570i	\(-0.157035\pi\)
\(19\)	−7.55436	−	0.793995i	−1.73309	−	0.182155i	−0.815250	−	0.579109i	\(-0.803401\pi\)
							−0.917839	+	0.396954i	\(0.870067\pi\)
\(23\)	4.89594	+	2.17981i	1.02087	+	0.454522i	0.847760	−	0.530380i	\(-0.177951\pi\)
							0.173114	+	0.984902i	\(0.444617\pi\)
\(29\)	9.68923	+	3.14822i	1.79924	+	0.584610i	0.999868	−	0.0162210i	\(-0.00516353\pi\)
							0.799376	+	0.600831i	\(0.205164\pi\)
\(31\)	8.35746	+	1.77643i	1.50104	+	0.319057i	0.883856	−	0.467759i	\(-0.154939\pi\)
							0.617188	+	0.786816i	\(0.288272\pi\)
\(37\)	0.316448	−	0.0672632i	0.0520238	−	0.0110580i	−0.181826	−	0.983331i	\(-0.558201\pi\)
							0.233850	+	0.972273i	\(0.424868\pi\)
\(41\)	0.0532268	−	6.40290i	0.00831263	−	0.999965i
							\(43\)	−3.56900	+	2.59303i	−0.544268	+	0.395434i	−0.825668	−	0.564157i	\(-0.809201\pi\)
0.281400	+	0.959591i	\(0.409201\pi\)
\(47\)	2.90917	−	6.53411i	0.424347	−	0.953098i	−0.567228	−	0.823561i	\(-0.691984\pi\)
							0.991575	−	0.129537i	\(-0.0413492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.19	✓	208
7.2	even	3	inner	287.2.z.a.86.8	yes	208
41.31	even	10	inner	287.2.z.a.277.8	yes	208
287.72	even	30	inner	287.2.z.a.72.19	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.19	✓	208	1.1	even	1	trivial
287.2.z.a.72.19	yes	208	287.72	even	30	inner
287.2.z.a.86.8	yes	208	7.2	even	3	inner
287.2.z.a.277.8	yes	208	41.31	even	10	inner