Embedded newform 287.2.r.c.9.15

• Label: 287.2.r.c.9.15
• 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			9.15
Character	\(\chi\)	\(=\)	287.9
Dual form			287.2.r.c.32.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.628687 + 0.362972i) q^{2} +(-0.172314 + 0.643084i) q^{3} +(-0.736502 - 1.27566i) q^{4} +(0.0766403 + 0.0442483i) q^{5} +(-0.341753 + 0.341753i) q^{6} +(0.877316 - 2.49606i) q^{7} -2.52121i q^{8} +(2.21421 + 1.27838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 4 q^{3} + 48 q^{4} - 28 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.628687	+	0.362972i	0.444549	+	0.256660i	0.705525	−	0.708685i	\(-0.250711\pi\)
							−0.260976	+	0.965345i	\(0.584045\pi\)
\(3\)	−0.172314	+	0.643084i	−0.0994854	+	0.371285i	−0.997661	−	0.0683517i	\(-0.978226\pi\)
							0.898176	+	0.439636i	\(0.144893\pi\)
\(5\)	0.0766403	+	0.0442483i	0.0342746	+	0.0197884i	0.517039	−	0.855962i	\(-0.327034\pi\)
							−0.482765	+	0.875750i	\(0.660367\pi\)
\(7\)	0.877316	−	2.49606i	0.331594	−	0.943422i
							\(11\)	0.998228	−	3.72544i	0.300977	−	1.12326i	−0.635376	−	0.772203i	\(-0.719155\pi\)
0.936353	−	0.351059i	\(-0.114178\pi\)
\(13\)	2.45847	+	2.45847i	0.681856	+	0.681856i	0.960418	−	0.278562i	\(-0.0898577\pi\)
							−0.278562	+	0.960418i	\(0.589858\pi\)
\(17\)	1.23818	+	0.331770i	0.300303	+	0.0804660i	0.405824	−	0.913951i	\(-0.366984\pi\)
							−0.105521	+	0.994417i	\(0.533651\pi\)
\(19\)	0.120612	+	0.450132i	0.0276704	+	0.103267i	0.978380	−	0.206815i	\(-0.0663099\pi\)
							−0.950710	+	0.310083i	\(0.899643\pi\)
\(23\)	−0.953848	+	1.65211i	−0.198891	+	0.344490i	−0.948169	−	0.317766i	\(-0.897067\pi\)
							0.749278	+	0.662256i	\(0.230401\pi\)
\(29\)	1.37390	+	1.37390i	0.255126	+	0.255126i	0.823068	−	0.567942i	\(-0.192260\pi\)
							−0.567942	+	0.823068i	\(0.692260\pi\)
\(31\)	−1.89718	−	3.28602i	−0.340744	−	0.590186i	0.643827	−	0.765171i	\(-0.277346\pi\)
							−0.984571	+	0.174985i	\(0.944012\pi\)
\(37\)	0.199257	−	0.345123i	0.0327576	−	0.0567378i	−0.849182	−	0.528101i	\(-0.822904\pi\)
							0.881939	+	0.471363i	\(0.156238\pi\)
\(41\)	0.568649	+	6.37782i	0.0888081	+	0.996049i
							\(43\)		−	1.39932i		−	0.213395i	−0.994292	−	0.106697i	\(-0.965972\pi\)
0.994292	−	0.106697i	\(-0.0340276\pi\)
\(47\)	0.180841	+	0.674908i	0.0263784	+	0.0984455i	0.977860	−	0.209260i	\(-0.0671055\pi\)
							−0.951482	+	0.307706i	\(0.900439\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.r.c.9.15	✓	96
7.4	even	3	inner	287.2.r.c.214.10	yes	96
41.32	even	4	inner	287.2.r.c.114.10	yes	96
287.32	even	12	inner	287.2.r.c.32.15	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.r.c.9.15	✓	96	1.1	even	1	trivial
287.2.r.c.32.15	yes	96	287.32	even	12	inner
287.2.r.c.114.10	yes	96	41.32	even	4	inner
287.2.r.c.214.10	yes	96	7.4	even	3	inner