Embedded newform 287.2.j.a.81.4

• Label: 287.2.j.a.81.4
• Level: $287$
• Weight: $2$
• Character: 287.81
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			81.4
Character	\(\chi\)	\(=\)	287.81
Dual form			287.2.j.a.163.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10471 - 1.91341i) q^{2} +(2.73490 + 1.57900i) q^{3} +(-1.44075 + 2.49545i) q^{4} +(1.43765 + 2.49009i) q^{5} -6.97730i q^{6} +(-2.25170 + 1.38919i) q^{7} +1.94760 q^{8} +(3.48645 + 6.03872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 2 q^{2} - 26 q^{4} - 6 q^{5} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10471	−	1.91341i	−0.781145	−	1.35298i	−0.931275	−	0.364317i	\(-0.881303\pi\)
							0.150130	−	0.988666i	\(-0.452031\pi\)
\(3\)	2.73490	+	1.57900i	1.57900	+	0.911634i	0.995000	+	0.0998768i	\(0.0318449\pi\)
							0.583996	+	0.811757i	\(0.301488\pi\)
\(5\)	1.43765	+	2.49009i	0.642937	+	1.11360i	0.984774	+	0.173841i	\(0.0556178\pi\)
							−0.341836	+	0.939759i	\(0.611049\pi\)
\(7\)	−2.25170	+	1.38919i	−0.851061	+	0.525066i
							\(11\)	−2.54549	−	1.46964i	−0.767493	−	0.443112i	0.0644864	−	0.997919i	\(-0.479459\pi\)
−0.831980	+	0.554806i	\(0.812792\pi\)
\(13\)		−	2.13267i		−	0.591497i	−0.955266	−	0.295749i	\(-0.904431\pi\)
							0.955266	−	0.295749i	\(-0.0955691\pi\)
\(17\)	0.886674	+	0.511922i	0.215050	+	0.124159i	0.603656	−	0.797245i	\(-0.293710\pi\)
							−0.388606	+	0.921404i	\(0.627043\pi\)
\(19\)	2.94895	−	1.70257i	0.676534	−	0.390597i	−0.122014	−	0.992528i	\(-0.538935\pi\)
							0.798548	+	0.601931i	\(0.205602\pi\)
\(23\)	2.74844	+	4.76043i	0.573089	+	0.992618i	0.996246	+	0.0865625i	\(0.0275882\pi\)
							−0.423158	+	0.906056i	\(0.639078\pi\)
\(29\)		−	4.32591i		−	0.803301i	−0.915793	−	0.401650i	\(-0.868437\pi\)
							0.915793	−	0.401650i	\(-0.131563\pi\)
\(31\)	3.84405	−	6.65810i	0.690413	−	1.19583i	−0.281290	−	0.959623i	\(-0.590762\pi\)
							0.971703	−	0.236207i	\(-0.0759043\pi\)
\(37\)	−3.04087	−	5.26693i	−0.499915	−	0.865878i	0.500085	−	0.865976i	\(-0.333302\pi\)
							−1.00000		9.79796e-5i	\(0.999969\pi\)
\(41\)	−3.85347	−	5.11379i	−0.601810	−	0.798639i
							\(43\)	10.5726			1.61231			0.806153	−	0.591707i	\(-0.201546\pi\)
0.806153	+	0.591707i	\(0.201546\pi\)
\(47\)	−2.25619	+	1.30261i	−0.329100	+	0.190006i	−0.655441	−	0.755246i	\(-0.727517\pi\)
							0.326342	+	0.945252i	\(0.394184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.j.a.81.4	yes	52
7.2	even	3	inner	287.2.j.a.163.3	yes	52
41.40	even	2	inner	287.2.j.a.81.3	✓	52
287.163	even	6	inner	287.2.j.a.163.4	yes	52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.j.a.81.3	✓	52	41.40	even	2	inner
287.2.j.a.81.4	yes	52	1.1	even	1	trivial
287.2.j.a.163.3	yes	52	7.2	even	3	inner
287.2.j.a.163.4	yes	52	287.163	even	6	inner