Embedded newform 288.2.bf.a.59.5

• Label: 288.2.bf.a.59.5
• Level: $288$
• Weight: $2$
• Character: 288.59
• 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			59.5
Character	\(\chi\)	\(=\)	288.59
Dual form			288.2.bf.a.83.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33382 + 0.470017i) q^{2} +(-1.03282 - 1.39043i) q^{3} +(1.55817 - 1.25384i) q^{4} +(2.03403 + 2.65080i) q^{5} +(2.03112 + 1.36915i) q^{6} +(-2.13389 - 0.571774i) q^{7} +(-1.48899 + 2.40476i) q^{8} +(-0.866584 + 2.87211i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 12 q^{2} - 8 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33382	+	0.470017i	−0.943155	+	0.332352i
							\(3\)	−1.03282	−	1.39043i	−0.596296	−	0.802764i
\(5\)	2.03403	+	2.65080i	0.909646	+	1.18547i								0.982149	+	0.188105i	\(0.0602347\pi\)
							−0.0725029	+	0.997368i	\(0.523099\pi\)
\(7\)	−2.13389	−	0.571774i	−0.806535	−	0.216110i	−0.168084	−	0.985773i	\(-0.553758\pi\)
							−0.638451	+	0.769662i	\(0.720425\pi\)
\(11\)	1.69328	−	0.222924i	0.510542	−	0.0672141i	0.129145	−	0.991626i	\(-0.458777\pi\)
							0.381396	+	0.924412i	\(0.375443\pi\)
\(13\)	−0.378852	+	2.87766i	−0.105075	+	0.798120i	0.855050	+	0.518546i	\(0.173527\pi\)
							−0.960124	+	0.279574i	\(0.909807\pi\)
\(17\)	2.32818			0.564666			0.282333	−	0.959316i	\(-0.408892\pi\)
							0.282333	+	0.959316i	\(0.408892\pi\)
\(19\)	0.0657086	+	0.158635i	0.0150746	+	0.0363933i	0.931238	−	0.364413i	\(-0.118730\pi\)
							−0.916163	+	0.400806i	\(0.868730\pi\)
\(23\)	2.45390	+	9.15808i	0.511674	+	1.90959i	0.402046	+	0.915620i	\(0.368299\pi\)
							0.109628	+	0.993973i	\(0.465034\pi\)
\(29\)	1.01919	+	0.782049i	0.189258	+	0.145223i	0.699045	−	0.715078i	\(-0.253609\pi\)
							−0.509786	+	0.860301i	\(0.670276\pi\)
\(31\)	4.08458	+	2.35823i	0.733613	+	0.423551i	0.819742	−	0.572733i	\(-0.194117\pi\)
							−0.0861297	+	0.996284i	\(0.527450\pi\)
\(37\)	1.93580	+	0.801833i	0.318243	+	0.131821i	0.536086	−	0.844163i	\(-0.319902\pi\)
							−0.217843	+	0.975984i	\(0.569902\pi\)
\(41\)	10.1796	−	2.72763i	1.58979	−	0.425984i	0.647854	−	0.761765i	\(-0.275667\pi\)
							0.941940	+	0.335781i	\(0.109000\pi\)
\(43\)	−0.123970	−	0.941643i	−0.0189052	−	0.143599i	0.979389	−	0.201983i	\(-0.0647386\pi\)
							−0.998294	+	0.0583839i	\(0.981405\pi\)
\(47\)	−8.34784	+	4.81963i	−1.21766	+	0.703015i	−0.964416	−	0.264388i	\(-0.914830\pi\)
							−0.253241	+	0.967403i	\(0.581497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.2.bf.a.59.5	✓	368
3.2	odd	2		864.2.bn.a.827.42		368
9.2	odd	6	inner	288.2.bf.a.155.12	yes	368
9.7	even	3		864.2.bn.a.251.35		368
32.19	odd	8	inner	288.2.bf.a.275.12	yes	368
96.83	even	8		864.2.bn.a.179.35		368
288.83	even	24	inner	288.2.bf.a.83.5	yes	368
288.115	odd	24		864.2.bn.a.467.42		368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.bf.a.59.5	✓	368	1.1	even	1	trivial
288.2.bf.a.83.5	yes	368	288.83	even	24	inner
288.2.bf.a.155.12	yes	368	9.2	odd	6	inner
288.2.bf.a.275.12	yes	368	32.19	odd	8	inner
864.2.bn.a.179.35		368	96.83	even	8
864.2.bn.a.251.35		368	9.7	even	3
864.2.bn.a.467.42		368	288.115	odd	24
864.2.bn.a.827.42		368	3.2	odd	2