Embedded newform 576.2.bm.a.205.20

• Label: 576.2.bm.a.205.20
• 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			205.20
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15985 + 0.809170i) q^{2} +(-0.561110 + 1.63864i) q^{3} +(0.690487 - 1.87703i) q^{4} +(-0.301585 - 0.343892i) q^{5} +(-0.675140 - 2.35461i) q^{6} +(0.560309 + 0.0737661i) q^{7} +(0.717975 + 2.73578i) q^{8} +(-2.37031 - 1.83892i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15985	+	0.809170i	−0.820135	+	0.572170i
							\(3\)	−0.561110	+	1.63864i	−0.323957	+	0.946072i
\(5\)	−0.301585	−	0.343892i	−0.134873	−	0.153793i								0.680461	−	0.732785i	\(-0.261780\pi\)
							−0.815334	+	0.578992i	\(0.803446\pi\)
\(7\)	0.560309	+	0.0737661i	0.211777	+	0.0278810i	0.235668	−	0.971834i	\(-0.424272\pi\)
							−0.0238911	+	0.999715i	\(0.507605\pi\)
\(11\)	−1.45150	−	2.94335i	−0.437644	−	0.887455i	−0.998236	−	0.0593628i	\(-0.981093\pi\)
							0.560592	−	0.828092i	\(-0.310574\pi\)
\(13\)	0.878865	+	2.58905i	0.243753	+	0.718074i	0.998101	+	0.0615989i	\(0.0196200\pi\)
							−0.754348	+	0.656475i	\(0.772047\pi\)
\(17\)	−5.57937	−	5.57937i	−1.35320	−	1.35320i	−0.882061	−	0.471135i	\(-0.843845\pi\)
							−0.471135	−	0.882061i	\(-0.656155\pi\)
\(19\)	−0.176008	+	0.884852i	−0.0403790	+	0.202999i	−0.995709	−	0.0925434i	\(-0.970500\pi\)
							0.955330	+	0.295542i	\(0.0955003\pi\)
\(23\)	−0.338768	−	2.57320i	−0.0706380	−	0.536549i	−0.989778	−	0.142618i	\(-0.954448\pi\)
							0.919140	−	0.393931i	\(-0.128885\pi\)
\(29\)	−8.99410	+	0.589505i	−1.67016	+	0.109468i	−0.870883	−	0.491490i	\(-0.836452\pi\)
							−0.799280	+	0.600959i	\(0.794786\pi\)
\(31\)	4.45142	−	2.57003i	0.799499	−	0.461591i	−0.0437972	−	0.999040i	\(-0.513946\pi\)
							0.843296	+	0.537450i	\(0.180612\pi\)
\(37\)	−1.00958	−	5.07551i	−0.165974	−	0.834409i	−0.970613	−	0.240645i	\(-0.922641\pi\)
							0.804639	−	0.593764i	\(-0.202359\pi\)
\(41\)	1.11839	+	8.49501i	0.174663	+	1.32670i	0.824170	+	0.566342i	\(0.191642\pi\)
							−0.649507	+	0.760356i	\(0.725025\pi\)
\(43\)	0.681764	−	0.336209i	0.103968	−	0.0512713i	−0.389561	−	0.921001i	\(-0.627373\pi\)
							0.493529	+	0.869729i	\(0.335707\pi\)
\(47\)	5.21246	−	1.39668i	0.760316	−	0.203726i	0.142227	−	0.989834i	\(-0.454574\pi\)
							0.618089	+	0.786108i	\(0.287907\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.20	yes	1504
9.4	even	3	inner	576.2.bm.a.13.44	✓	1504
64.5	even	16	inner	576.2.bm.a.133.44	yes	1504
576.517	even	48	inner	576.2.bm.a.517.20	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.44	✓	1504	9.4	even	3	inner
576.2.bm.a.133.44	yes	1504	64.5	even	16	inner
576.2.bm.a.205.20	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.20	yes	1504	576.517	even	48	inner