Embedded newform 576.2.bm.a.205.80

• Label: 576.2.bm.a.205.80
• Level: $576$
• Weight: $2$
• Character: 576.205
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $1504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.bm (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(1504\)
Relative dimension:	\(94\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			205.80
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.80

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.26096 + 0.640304i) q^{2} +(1.37357 + 1.05513i) q^{3} +(1.18002 + 1.61479i) q^{4} +(0.347943 + 0.396753i) q^{5} +(1.05640 + 2.20998i) q^{6} +(2.56782 + 0.338060i) q^{7} +(0.454001 + 2.79175i) q^{8} +(0.773384 + 2.89860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.26096	+	0.640304i	0.891631	+	0.452763i
							\(3\)	1.37357	+	1.05513i	0.793030	+	0.609182i
\(5\)	0.347943	+	0.396753i	0.155605	+	0.177434i								0.824390	−	0.566023i	\(-0.191519\pi\)
							−0.668785	+	0.743456i	\(0.733185\pi\)
\(7\)	2.56782	+	0.338060i	0.970544	+	0.127775i	0.599094	−	0.800679i	\(-0.295527\pi\)
							0.371450	+	0.928453i	\(0.378861\pi\)
\(11\)	−1.93917	−	3.93224i	−0.584681	−	1.18562i	−0.965363	−	0.260910i	\(-0.915977\pi\)
							0.380682	−	0.924706i	\(-0.375689\pi\)
\(13\)	−1.46592	−	4.31846i	−0.406573	−	1.19773i	−0.936799	−	0.349869i	\(-0.886226\pi\)
							0.530225	−	0.847857i	\(-0.322107\pi\)
\(17\)	−3.42677	−	3.42677i	−0.831113	−	0.831113i	0.156556	−	0.987669i	\(-0.449961\pi\)
							−0.987669	+	0.156556i	\(0.949961\pi\)
\(19\)	−0.675985	+	3.39840i	−0.155082	+	0.779647i	0.822447	+	0.568842i	\(0.192609\pi\)
							−0.977528	+	0.210805i	\(0.932391\pi\)
\(23\)	−0.716559	−	5.44281i	−0.149413	−	1.13490i	−0.887702	−	0.460419i	\(-0.847699\pi\)
							0.738289	−	0.674485i	\(-0.235634\pi\)
\(29\)	−2.67205	+	0.175136i	−0.496188	+	0.0325219i	−0.311443	−	0.950265i	\(-0.600812\pi\)
							−0.184744	+	0.982787i	\(0.559146\pi\)
\(31\)	−3.23242	+	1.86624i	−0.580560	+	0.335187i	−0.761356	−	0.648334i	\(-0.775466\pi\)
							0.180796	+	0.983521i	\(0.442133\pi\)
\(37\)	1.61701	+	8.12927i	0.265835	+	1.33644i	0.850845	+	0.525418i	\(0.176091\pi\)
							−0.585009	+	0.811027i	\(0.698909\pi\)
\(41\)	1.30537	+	9.91529i	0.203865	+	1.54851i	0.721201	+	0.692726i	\(0.243590\pi\)
							−0.517336	+	0.855782i	\(0.673076\pi\)
\(43\)	4.70955	−	2.32250i	0.718200	−	0.354177i	−0.0461885	−	0.998933i	\(-0.514707\pi\)
							0.764389	+	0.644756i	\(0.223041\pi\)
\(47\)	6.57313	−	1.76127i	0.958790	−	0.256907i	0.254702	−	0.967020i	\(-0.418023\pi\)
							0.704088	+	0.710113i	\(0.251356\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.80	yes	1504
9.4	even	3	inner	576.2.bm.a.13.18	✓	1504
64.5	even	16	inner	576.2.bm.a.133.18	yes	1504
576.517	even	48	inner	576.2.bm.a.517.80	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.18	✓	1504	9.4	even	3	inner
576.2.bm.a.133.18	yes	1504	64.5	even	16	inner
576.2.bm.a.205.80	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.80	yes	1504	576.517	even	48	inner