Embedded newform 288.2.bf.a.11.3

• Label: 288.2.bf.a.11.3
• Level: $288$
• Weight: $2$
• Character: 288.11
• Analytic conductor: $2.300$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.bf (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29969157821\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			11.3
Character	\(\chi\)	\(=\)	288.11
Dual form			288.2.bf.a.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40115 + 0.191775i) q^{2} +(-1.72586 + 0.146277i) q^{3} +(1.92644 - 0.537410i) q^{4} +(0.234629 + 1.78218i) q^{5} +(2.39014 - 0.535934i) q^{6} +(-0.927052 - 3.45980i) q^{7} +(-2.59618 + 1.12244i) q^{8} +(2.95721 - 0.504910i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40115	+	0.191775i	−0.990763	+	0.135605i
							\(3\)	−1.72586	+	0.146277i	−0.996427	+	0.0844533i
\(5\)	0.234629	+	1.78218i	0.104929	+	0.797017i								0.960294	+	0.278991i	\(0.0899998\pi\)
							−0.855365	+	0.518027i	\(0.826667\pi\)
\(7\)	−0.927052	−	3.45980i	−0.350393	−	1.30768i	−0.886184	−	0.463333i	\(-0.846653\pi\)
							0.535792	−	0.844350i	\(-0.320013\pi\)
\(11\)	0.296127	−	0.227226i	0.0892857	−	0.0685113i	−0.563154	−	0.826352i	\(-0.690412\pi\)
							0.652440	+	0.757841i	\(0.273746\pi\)
\(13\)	−1.82398	+	2.37705i	−0.505880	+	0.659276i	−0.974569	−	0.224088i	\(-0.928060\pi\)
							0.468689	+	0.883363i	\(0.344726\pi\)
\(17\)	1.74490			0.423201			0.211601	−	0.977356i	\(-0.432132\pi\)
							0.211601	+	0.977356i	\(0.432132\pi\)
\(19\)	5.20446	−	2.15576i	1.19398	−	0.494565i	0.304934	−	0.952374i	\(-0.401366\pi\)
							0.889051	+	0.457809i	\(0.151366\pi\)
\(23\)	6.63081	+	1.77672i	1.38262	+	0.370472i	0.872072	−	0.489378i	\(-0.162776\pi\)
							0.510547	+	0.859850i	\(0.329443\pi\)
\(29\)	5.67339	+	0.746916i	1.05352	+	0.138699i	0.637340	−	0.770583i	\(-0.280035\pi\)
							0.416182	+	0.909281i	\(0.363368\pi\)
\(31\)	2.71370	−	1.56676i	0.487395	−	0.281398i	−0.236098	−	0.971729i	\(-0.575869\pi\)
							0.723493	+	0.690331i	\(0.242535\pi\)
\(37\)	0.845101	−	2.04026i	0.138934	−	0.335416i	−0.839063	−	0.544034i	\(-0.816896\pi\)
							0.977997	+	0.208618i	\(0.0668964\pi\)
\(41\)	0.247421	−	0.923388i	0.0386407	−	0.144209i	−0.943910	−	0.330201i	\(-0.892883\pi\)
							0.982551	+	0.185992i	\(0.0595500\pi\)
\(43\)	7.07675	+	9.22260i	1.07919	+	1.40643i	0.906879	+	0.421391i	\(0.138458\pi\)
							0.172314	+	0.985042i	\(0.444876\pi\)
\(47\)	−6.77779	−	3.91316i	−0.988643	−	0.570793i	−0.0837745	−	0.996485i	\(-0.526698\pi\)
							−0.904868	+	0.425692i	\(0.860031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.2.bf.a.11.3	✓	368
3.2	odd	2		864.2.bn.a.683.44		368
9.4	even	3		864.2.bn.a.395.27		368
9.5	odd	6	inner	288.2.bf.a.203.20	yes	368
32.3	odd	8	inner	288.2.bf.a.227.20	yes	368
96.35	even	8		864.2.bn.a.35.27		368
288.67	odd	24		864.2.bn.a.611.44		368
288.131	even	24	inner	288.2.bf.a.131.3	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.bf.a.11.3	✓	368	1.1	even	1	trivial
288.2.bf.a.131.3	yes	368	288.131	even	24	inner
288.2.bf.a.203.20	yes	368	9.5	odd	6	inner
288.2.bf.a.227.20	yes	368	32.3	odd	8	inner
864.2.bn.a.35.27		368	96.35	even	8
864.2.bn.a.395.27		368	9.4	even	3
864.2.bn.a.611.44		368	288.67	odd	24
864.2.bn.a.683.44		368	3.2	odd	2