Embedded newform 576.2.bm.a.205.10

• Label: 576.2.bm.a.205.10
• 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			205.10
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35510 + 0.404590i) q^{2} +(0.578280 + 1.63266i) q^{3} +(1.67261 - 1.09652i) q^{4} +(-1.08161 - 1.23334i) q^{5} +(-1.44419 - 1.97846i) q^{6} +(-2.69081 - 0.354252i) q^{7} +(-1.82292 + 2.16263i) q^{8} +(-2.33118 + 1.88827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35510	+	0.404590i	−0.958203	+	0.286089i
							\(3\)	0.578280	+	1.63266i	0.333870	+	0.942619i
\(5\)	−1.08161	−	1.23334i	−0.483712	−	0.551568i								0.457577	−	0.889170i	\(-0.348717\pi\)
							−0.941289	+	0.337602i	\(0.890384\pi\)
\(7\)	−2.69081	−	0.354252i	−1.01703	−	0.133895i	−0.396466	−	0.918049i	\(-0.629764\pi\)
							−0.620566	+	0.784155i	\(0.713097\pi\)
\(11\)	1.67332	+	3.39317i	0.504526	+	1.02308i	0.988951	+	0.148245i	\(0.0473625\pi\)
							−0.484424	+	0.874833i	\(0.660971\pi\)
\(13\)	−2.14523	−	6.31963i	−0.594979	−	1.75275i	−0.656158	−	0.754623i	\(-0.727820\pi\)
							0.0611796	−	0.998127i	\(-0.480514\pi\)
\(17\)	1.12973	+	1.12973i	0.274000	+	0.274000i	0.830708	−	0.556708i	\(-0.187936\pi\)
							−0.556708	+	0.830708i	\(0.687936\pi\)
\(19\)	0.200277	−	1.00686i	0.0459468	−	0.230990i	−0.950988	−	0.309228i	\(-0.899929\pi\)
							0.996935	+	0.0782380i	\(0.0249294\pi\)
\(23\)	−0.716717	−	5.44401i	−0.149446	−	1.13515i	−0.887631	−	0.460556i	\(-0.847650\pi\)
							0.738185	−	0.674599i	\(-0.235683\pi\)
\(29\)	−3.39404	+	0.222457i	−0.630258	+	0.0413093i	−0.377185	−	0.926138i	\(-0.623108\pi\)
							−0.253073	+	0.967447i	\(0.581441\pi\)
\(31\)	−3.29701	+	1.90353i	−0.592160	+	0.341884i	−0.765951	−	0.642899i	\(-0.777732\pi\)
							0.173791	+	0.984783i	\(0.444398\pi\)
\(37\)	−1.66443	−	8.36765i	−0.273630	−	1.37563i	−0.835992	−	0.548742i	\(-0.815107\pi\)
							0.562361	−	0.826892i	\(-0.309893\pi\)
\(41\)	−0.893620	−	6.78772i	−0.139560	−	1.06006i	−0.907652	−	0.419724i	\(-0.862127\pi\)
							0.768092	−	0.640340i	\(-0.221206\pi\)
\(43\)	4.18302	−	2.06284i	0.637905	−	0.314580i	−0.0944230	−	0.995532i	\(-0.530101\pi\)
							0.732328	+	0.680952i	\(0.238434\pi\)
\(47\)	2.15714	−	0.578003i	0.314651	−	0.0843104i	−0.0980376	−	0.995183i	\(-0.531257\pi\)
							0.412688	+	0.910872i	\(0.364590\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.10	yes	1504
9.4	even	3	inner	576.2.bm.a.13.56	✓	1504
64.5	even	16	inner	576.2.bm.a.133.56	yes	1504
576.517	even	48	inner	576.2.bm.a.517.10	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.56	✓	1504	9.4	even	3	inner
576.2.bm.a.133.56	yes	1504	64.5	even	16	inner
576.2.bm.a.205.10	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.10	yes	1504	576.517	even	48	inner