Embedded newform 576.2.bm.a.13.9

• Label: 576.2.bm.a.13.9
• Level: $576$
• Weight: $2$
• Character: 576.13
• 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			13.9
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35331 - 0.410562i) q^{2} +(-0.328391 - 1.70063i) q^{3} +(1.66288 + 1.11123i) q^{4} +(0.981205 - 2.89054i) q^{5} +(-0.253802 + 2.43631i) q^{6} +(0.185279 - 0.241461i) q^{7} +(-1.79416 - 2.18655i) q^{8} +(-2.78432 + 1.11695i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35331	−	0.410562i	−0.956932	−	0.290311i
							\(3\)	−0.328391	−	1.70063i	−0.189597	−	0.981862i
\(5\)	0.981205	−	2.89054i	0.438808	−	1.29269i								−0.472634	−	0.881259i	\(-0.656697\pi\)
							0.911442	−	0.411428i	\(-0.134970\pi\)
\(7\)	0.185279	−	0.241461i	0.0700290	−	0.0912636i	−0.757025	−	0.653385i	\(-0.773348\pi\)
							0.827054	+	0.562122i	\(0.190015\pi\)
\(11\)	−4.41470	−	0.289355i	−1.33108	−	0.0872438i	−0.616811	−	0.787112i	\(-0.711576\pi\)
							−0.714272	+	0.699868i	\(0.753242\pi\)
\(13\)	−4.03888	+	4.60546i	−1.12018	+	1.27732i	−0.163389	+	0.986562i	\(0.552242\pi\)
							−0.956795	+	0.290763i	\(0.906091\pi\)
\(17\)	0.270933	+	0.270933i	0.0657110	+	0.0657110i	0.739199	−	0.673488i	\(-0.235205\pi\)
							−0.673488	+	0.739199i	\(0.735205\pi\)
\(19\)	0.736341	−	3.70184i	0.168928	−	0.849260i	−0.799634	−	0.600488i	\(-0.794973\pi\)
							0.968562	−	0.248772i	\(-0.0800269\pi\)
\(23\)	−0.326665	+	0.250659i	−0.0681144	+	0.0522661i	−0.642257	−	0.766489i	\(-0.722002\pi\)
							0.574143	+	0.818755i	\(0.305335\pi\)
\(29\)	2.51339	−	5.09666i	0.466726	−	0.946426i	−0.528510	−	0.848927i	\(-0.677249\pi\)
							0.995236	−	0.0974989i	\(-0.0310842\pi\)
\(31\)	−6.44397	−	3.72043i	−1.15737	−	0.668209i	−0.206699	−	0.978405i	\(-0.566272\pi\)
							−0.950673	+	0.310196i	\(0.899605\pi\)
\(37\)	0.821026	+	4.12758i	0.134976	+	0.678569i	0.987719	+	0.156239i	\(0.0499371\pi\)
							−0.852743	+	0.522330i	\(0.825063\pi\)
\(41\)	−5.19055	+	3.98285i	−0.810627	+	0.622016i	−0.928634	−	0.370997i	\(-0.879016\pi\)
							0.118007	+	0.993013i	\(0.462349\pi\)
\(43\)	0.542590	−	8.27833i	0.0827442	−	1.26243i	−0.731472	−	0.681871i	\(-0.761166\pi\)
							0.814216	−	0.580561i	\(-0.197167\pi\)
\(47\)	−1.80514	+	6.73689i	−0.263307	+	0.982676i	0.699971	+	0.714171i	\(0.253196\pi\)
							−0.963278	+	0.268505i	\(0.913471\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.9	✓	1504
9.7	even	3	inner	576.2.bm.a.205.53	yes	1504
64.5	even	16	inner	576.2.bm.a.517.53	yes	1504
576.133	even	48	inner	576.2.bm.a.133.9	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.9	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.9	yes	1504	576.133	even	48	inner
576.2.bm.a.205.53	yes	1504	9.7	even	3	inner
576.2.bm.a.517.53	yes	1504	64.5	even	16	inner