Embedded newform 576.2.bm.a.205.94

• Label: 576.2.bm.a.205.94
• 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			205.94
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.94

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41400 - 0.0248533i) q^{2} +(-1.33031 + 1.10917i) q^{3} +(1.99876 - 0.0702848i) q^{4} +(0.404828 + 0.461618i) q^{5} +(-1.85349 + 1.60143i) q^{6} +(2.92960 + 0.385690i) q^{7} +(2.82450 - 0.149058i) q^{8} +(0.539468 - 2.95110i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41400	−	0.0248533i	0.999846	−	0.0175739i
							\(3\)	−1.33031	+	1.10917i	−0.768057	+	0.640382i
\(5\)	0.404828	+	0.461618i	0.181044	+	0.206442i								0.835243	−	0.549881i	\(-0.185327\pi\)
							−0.654198	+	0.756323i	\(0.726994\pi\)
\(7\)	2.92960	+	0.385690i	1.10729	+	0.145777i	0.661922	−	0.749573i	\(-0.269741\pi\)
							0.445364	+	0.895350i	\(0.353074\pi\)
\(11\)	−1.36139	−	2.76063i	−0.410475	−	0.832361i	−0.999702	−	0.0243935i	\(-0.992235\pi\)
							0.589228	−	0.807967i	\(-0.299432\pi\)
\(13\)	1.93357	+	5.69610i	0.536275	+	1.57981i	0.790367	+	0.612634i	\(0.209890\pi\)
							−0.254092	+	0.967180i	\(0.581777\pi\)
\(17\)	−3.43323	−	3.43323i	−0.832681	−	0.832681i	0.155201	−	0.987883i	\(-0.450397\pi\)
							−0.987883	+	0.155201i	\(0.950397\pi\)
\(19\)	−0.644204	+	3.23863i	−0.147791	+	0.742993i	0.833811	+	0.552051i	\(0.186154\pi\)
							−0.981601	+	0.190943i	\(0.938846\pi\)
\(23\)	0.364355	+	2.76755i	0.0759734	+	0.577075i	0.986517	+	0.163661i	\(0.0523304\pi\)
							−0.910543	+	0.413414i	\(0.864336\pi\)
\(29\)	10.5801	−	0.693457i	1.96468	−	0.128772i	0.970849	−	0.239692i	\(-0.0770466\pi\)
							0.993829	+	0.110920i	\(0.0353799\pi\)
\(31\)	−7.52420	+	4.34410i	−1.35139	+	0.780223i	−0.988444	−	0.151589i	\(-0.951561\pi\)
							−0.362942	+	0.931812i	\(0.618228\pi\)
\(37\)	0.774146	+	3.89190i	0.127269	+	0.639824i	0.990779	+	0.135492i	\(0.0432614\pi\)
							−0.863510	+	0.504332i	\(0.831739\pi\)
\(41\)	−0.251071	−	1.90707i	−0.0392106	−	0.297834i	−0.999803	−	0.0198718i	\(-0.993674\pi\)
							0.960592	−	0.277963i	\(-0.0896591\pi\)
\(43\)	−3.49148	+	1.72181i	−0.532446	+	0.262573i	−0.688577	−	0.725163i	\(-0.741764\pi\)
							0.156131	+	0.987736i	\(0.450098\pi\)
\(47\)	−4.57068	+	1.22471i	−0.666702	+	0.178642i	−0.576269	−	0.817260i	\(-0.695492\pi\)
							−0.0904330	+	0.995903i	\(0.528825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.94	yes	1504
9.4	even	3	inner	576.2.bm.a.13.33	✓	1504
64.5	even	16	inner	576.2.bm.a.133.33	yes	1504
576.517	even	48	inner	576.2.bm.a.517.94	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.33	✓	1504	9.4	even	3	inner
576.2.bm.a.133.33	yes	1504	64.5	even	16	inner
576.2.bm.a.205.94	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.94	yes	1504	576.517	even	48	inner