Embedded newform 288.2.bc.a.61.21

• Label: 288.2.bc.a.61.21
• Level: $288$
• Weight: $2$
• Character: 288.61
• 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.bc (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			61.21
Character	\(\chi\)	\(=\)	288.61
Dual form			288.2.bc.a.85.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.370884 - 1.36471i) q^{2} +(-1.35285 + 1.08157i) q^{3} +(-1.72489 + 1.01230i) q^{4} +(-0.404201 + 0.526765i) q^{5} +(1.97778 + 1.44512i) q^{6} +(-0.243276 - 0.907918i) q^{7} +(2.02124 + 1.97854i) q^{8} +(0.660425 - 2.92640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 4 q^{2} - 8 q^{3} - 4 q^{4} - 4 q^{5} - 8 q^{6} - 4 q^{7} - 16 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.370884	−	1.36471i	−0.262254	−	0.964999i
							\(3\)	−1.35285	+	1.08157i	−0.781070	+	0.624443i
\(5\)	−0.404201	+	0.526765i	−0.180764	+	0.235577i								−0.874943	−	0.484226i	\(-0.839101\pi\)
							0.694179	+	0.719803i	\(0.255768\pi\)
\(7\)	−0.243276	−	0.907918i	−0.0919496	−	0.343161i	0.904590	−	0.426283i	\(-0.140177\pi\)
							−0.996539	+	0.0831227i	\(0.973511\pi\)
\(11\)	0.421338	−	3.20038i	0.127038	−	0.964952i	−0.802610	−	0.596504i	\(-0.796556\pi\)
							0.929648	−	0.368448i	\(-0.120111\pi\)
\(13\)	4.19601	−	0.552416i	1.16376	−	0.153213i	0.476201	−	0.879337i	\(-0.342014\pi\)
							0.687564	+	0.726124i	\(0.258680\pi\)
\(17\)		−	7.66398i		−	1.85879i	−0.369088	−	0.929394i	\(-0.620330\pi\)
							0.369088	−	0.929394i	\(-0.379670\pi\)
\(19\)	−1.08544	+	2.62049i	−0.249018	+	0.601183i	−0.998121	−	0.0612707i	\(-0.980485\pi\)
							0.749103	+	0.662453i	\(0.230485\pi\)
\(23\)	0.554629	−	2.06990i	0.115648	−	0.431604i	−0.883687	−	0.468079i	\(-0.844946\pi\)
							0.999335	+	0.0364748i	\(0.0116129\pi\)
\(29\)	6.07554	−	4.66193i	1.12820	−	0.865698i	0.136043	−	0.990703i	\(-0.456561\pi\)
							0.992156	+	0.125005i	\(0.0398947\pi\)
\(31\)	−3.79029	−	6.56498i	−0.680756	−	1.17910i	−0.974750	−	0.223297i	\(-0.928318\pi\)
							0.293994	−	0.955807i	\(-0.405015\pi\)
\(37\)	0.843770	+	2.03704i	0.138715	+	0.334887i	0.977937	−	0.208902i	\(-0.0669890\pi\)
							−0.839222	+	0.543790i	\(0.816989\pi\)
\(41\)	−0.0918226	+	0.342687i	−0.0143403	+	0.0535187i	−0.972725	−	0.231960i	\(-0.925486\pi\)
							0.958385	+	0.285479i	\(0.0921527\pi\)
\(43\)	−0.690978	+	5.24850i	−0.105373	+	0.800389i	0.854401	+	0.519615i	\(0.173925\pi\)
							−0.959774	+	0.280774i	\(0.909409\pi\)
\(47\)	−4.75995	−	2.74816i	−0.694310	−	0.400860i	0.110915	−	0.993830i	\(-0.464622\pi\)
							−0.805225	+	0.592970i	\(0.797955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.2.bc.a.61.21	✓	368
3.2	odd	2		864.2.bk.a.829.26		368
9.4	even	3	inner	288.2.bc.a.157.42	yes	368
9.5	odd	6		864.2.bk.a.253.5		368
32.21	even	8	inner	288.2.bc.a.277.42	yes	368
96.53	odd	8		864.2.bk.a.181.5		368
288.85	even	24	inner	288.2.bc.a.85.21	yes	368
288.149	odd	24		864.2.bk.a.469.26		368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.bc.a.61.21	✓	368	1.1	even	1	trivial
288.2.bc.a.85.21	yes	368	288.85	even	24	inner
288.2.bc.a.157.42	yes	368	9.4	even	3	inner
288.2.bc.a.277.42	yes	368	32.21	even	8	inner
864.2.bk.a.181.5		368	96.53	odd	8
864.2.bk.a.253.5		368	9.5	odd	6
864.2.bk.a.469.26		368	288.149	odd	24
864.2.bk.a.829.26		368	3.2	odd	2