Embedded newform 576.2.bm.a.517.10

• Label: 576.2.bm.a.517.10
• 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			517.10
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.10

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.56	✓	1504	64.13	even	16	inner
576.2.bm.a.133.56	yes	1504	9.7	even	3	inner
576.2.bm.a.205.10	yes	1504	576.205	even	48	inner
576.2.bm.a.517.10	yes	1504	1.1	even	1	trivial