Embedded newform 287.2.z.a.4.6

• Label: 287.2.z.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	287.4
Dual form			287.2.z.a.72.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78806 - 0.796097i) q^{2} +(-2.45203 - 1.41568i) q^{3} +(1.22514 + 1.36066i) q^{4} +(-3.36959 + 0.716229i) q^{5} +(3.25737 + 4.48338i) q^{6} +(2.05104 + 1.67130i) q^{7} +(0.102251 + 0.314698i) q^{8} +(2.50830 + 4.34450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 208 q - 3 q^{2} + 21 q^{4} + q^{5} - 40 q^{6} - 10 q^{7} - 8 q^{8} + 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78806	−	0.796097i	−1.26435	−	0.562926i	−0.338555	−	0.940947i	\(-0.609938\pi\)
							−0.925797	+	0.378021i	\(0.876605\pi\)
\(3\)	−2.45203	−	1.41568i	−1.41568	−	0.817343i	−0.419764	−	0.907633i	\(-0.637887\pi\)
							−0.995915	+	0.0902904i	\(0.971220\pi\)
\(5\)	−3.36959	+	0.716229i	−1.50693	+	0.320308i	−0.886046	−	0.463597i	\(-0.846559\pi\)
							−0.620882	+	0.783904i	\(0.713225\pi\)
\(7\)	2.05104	+	1.67130i	0.775219	+	0.631693i
							\(11\)	−0.273910	+	1.28865i	−0.0825871	+	0.388542i	−0.999955	−	0.00949895i	\(-0.996976\pi\)
0.917368	+	0.398041i	\(0.130310\pi\)
\(13\)	−3.86136	−	5.31471i	−1.07095	−	1.47403i	−0.869112	−	0.494615i	\(-0.835309\pi\)
							−0.201837	−	0.979419i	\(-0.564691\pi\)
\(17\)	−0.751439	+	3.53524i	−0.182251	+	0.857423i	0.788067	+	0.615589i	\(0.211082\pi\)
							−0.970318	+	0.241833i	\(0.922251\pi\)
\(19\)	−0.616319	−	0.0647778i	−0.141393	−	0.0148610i	0.0335674	−	0.999436i	\(-0.489313\pi\)
							−0.174961	+	0.984575i	\(0.555980\pi\)
\(23\)	−5.53130	−	2.46269i	−1.15336	−	0.513507i	−0.261222	−	0.965279i	\(-0.584126\pi\)
							−0.892134	+	0.451771i	\(0.850792\pi\)
\(29\)	6.12381	+	1.98975i	1.13716	+	0.369486i	0.816294	−	0.577637i	\(-0.196025\pi\)
							0.320869	+	0.947124i	\(0.396025\pi\)
\(31\)	4.27978	+	0.909695i	0.768671	+	0.163386i	0.575525	−	0.817784i	\(-0.304798\pi\)
							0.193146	+	0.981170i	\(0.438131\pi\)
\(37\)	5.14974	−	1.09461i	0.846612	−	0.179953i	0.235872	−	0.971784i	\(-0.424205\pi\)
							0.610740	+	0.791831i	\(0.290872\pi\)
\(41\)	5.52945	−	3.22881i	0.863555	−	0.504255i
							\(43\)	3.40007	−	2.47030i	0.518506	−	0.376716i	−0.297535	−	0.954711i	\(-0.596164\pi\)
0.816041	+	0.577994i	\(0.196164\pi\)
\(47\)	3.41877	−	7.67867i	0.498678	−	1.12005i	−0.472421	−	0.881373i	\(-0.656620\pi\)
							0.971099	−	0.238676i	\(-0.0767135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.z.a.4.6	✓	208
7.2	even	3	inner	287.2.z.a.86.21	yes	208
41.31	even	10	inner	287.2.z.a.277.21	yes	208
287.72	even	30	inner	287.2.z.a.72.6	yes	208

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.z.a.4.6	✓	208	1.1	even	1	trivial
287.2.z.a.72.6	yes	208	287.72	even	30	inner
287.2.z.a.86.21	yes	208	7.2	even	3	inner
287.2.z.a.277.21	yes	208	41.31	even	10	inner