Embedded newform 576.2.bm.a.517.76

• Label: 576.2.bm.a.517.76
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.76
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.76

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09828 + 0.890946i) q^{2} +(0.224666 - 1.71742i) q^{3} +(0.412432 + 1.95701i) q^{4} +(2.72176 - 3.10357i) q^{5} +(1.77687 - 1.68604i) q^{6} +(2.35864 - 0.310521i) q^{7} +(-1.29063 + 2.51680i) q^{8} +(-2.89905 - 0.771692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09828	+	0.890946i	0.776600	+	0.629994i
							\(3\)	0.224666	−	1.71742i	0.129711	−	0.991552i
\(5\)	2.72176	−	3.10357i	1.21721	−	1.38796i								0.319373	−	0.947629i	\(-0.396528\pi\)
							0.897835	−	0.440331i	\(-0.145139\pi\)
\(7\)	2.35864	−	0.310521i	0.891483	−	0.117366i	0.329151	−	0.944277i	\(-0.393238\pi\)
							0.562332	+	0.826911i	\(0.309904\pi\)
\(11\)	0.107104	−	0.217184i	0.0322929	−	0.0654836i	−0.880128	−	0.474736i	\(-0.842544\pi\)
							0.912421	+	0.409252i	\(0.134210\pi\)
\(13\)	−0.686105	+	2.02120i	−0.190291	+	0.560580i	−0.999642	−	0.0267450i	\(-0.991486\pi\)
							0.809351	+	0.587325i	\(0.199819\pi\)
\(17\)	−4.79616	+	4.79616i	−1.16324	+	1.16324i	−0.179478	+	0.983762i	\(0.557441\pi\)
							−0.983762	+	0.179478i	\(0.942559\pi\)
\(19\)	−1.45921	−	7.33592i	−0.334765	−	1.68298i	−0.671206	−	0.741271i	\(-0.734223\pi\)
							0.336441	−	0.941705i	\(-0.390777\pi\)
\(23\)	−0.332854	+	2.52827i	−0.0694048	+	0.527182i	0.921061	+	0.389419i	\(0.127324\pi\)
							−0.990465	+	0.137762i	\(0.956009\pi\)
\(29\)	4.29270	+	0.281358i	0.797134	+	0.0522469i	0.458527	−	0.888681i	\(-0.348377\pi\)
							0.338608	+	0.940928i	\(0.390044\pi\)
\(31\)	7.75983	+	4.48014i	1.39371	+	0.804656i	0.993723	−	0.111866i	\(-0.0356828\pi\)
							0.399983	+	0.916523i	\(0.369016\pi\)
\(37\)	−0.502267	+	2.52507i	−0.0825722	+	0.415118i	0.917285	+	0.398231i	\(0.130376\pi\)
							−0.999857	+	0.0168876i	\(0.994624\pi\)
\(41\)	−0.940526	+	7.14400i	−0.146885	+	1.11571i	0.746169	+	0.665757i	\(0.231891\pi\)
							−0.893054	+	0.449949i	\(0.851442\pi\)
\(43\)	6.11085	+	3.01354i	0.931895	+	0.459560i	0.843864	−	0.536558i	\(-0.180276\pi\)
							0.0880317	+	0.996118i	\(0.471942\pi\)
\(47\)	−9.30764	−	2.49397i	−1.35766	−	0.363784i	−0.494703	−	0.869062i	\(-0.664723\pi\)
							−0.862956	+	0.505279i	\(0.831390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.76	yes	1504
9.7	even	3	inner	576.2.bm.a.133.52	yes	1504
64.13	even	16	inner	576.2.bm.a.13.52	✓	1504
576.205	even	48	inner	576.2.bm.a.205.76	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.52	✓	1504	64.13	even	16	inner
576.2.bm.a.133.52	yes	1504	9.7	even	3	inner
576.2.bm.a.205.76	yes	1504	576.205	even	48	inner
576.2.bm.a.517.76	yes	1504	1.1	even	1	trivial