Embedded newform 576.2.bm.a.205.29

• Label: 576.2.bm.a.205.29
• 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			205.29
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.832522 - 1.14320i) q^{2} +(-0.342098 - 1.69793i) q^{3} +(-0.613814 + 1.90348i) q^{4} +(1.06020 + 1.20893i) q^{5} +(-1.65627 + 1.80465i) q^{6} +(0.0200891 + 0.00264478i) q^{7} +(2.68707 - 0.882976i) q^{8} +(-2.76594 + 1.16172i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.832522	−	1.14320i	−0.588682	−	0.808365i
							\(3\)	−0.342098	−	1.69793i	−0.197510	−	0.980301i
\(5\)	1.06020	+	1.20893i	0.474136	+	0.540649i								0.938629	−	0.344928i	\(-0.112097\pi\)
							−0.464493	+	0.885577i	\(0.653763\pi\)
\(7\)	0.0200891	+	0.00264478i	0.00759297	+	0.000999634i	0.134322	−	0.990938i	\(-0.457114\pi\)
							−0.126729	+	0.991937i	\(0.540448\pi\)
\(11\)	0.486341	+	0.986202i	0.146637	+	0.297351i	0.957783	−	0.287494i	\(-0.0928220\pi\)
							−0.811145	+	0.584845i	\(0.801155\pi\)
\(13\)	−1.37041	−	4.03710i	−0.380084	−	1.11969i	−0.953750	−	0.300602i	\(-0.902813\pi\)
							0.573666	−	0.819089i	\(-0.305521\pi\)
\(17\)	−2.58058	−	2.58058i	−0.625882	−	0.625882i	0.321147	−	0.947029i	\(-0.395932\pi\)
							−0.947029	+	0.321147i	\(0.895932\pi\)
\(19\)	0.725210	−	3.64588i	0.166375	−	0.836421i	−0.803965	−	0.594676i	\(-0.797280\pi\)
							0.970340	−	0.241745i	\(-0.0777197\pi\)
\(23\)	−0.902673	−	6.85648i	−0.188220	−	1.42968i	−0.781111	−	0.624392i	\(-0.785347\pi\)
							0.592890	−	0.805283i	\(-0.297987\pi\)
\(29\)	−8.38882	+	0.549832i	−1.55776	+	0.102101i	−0.819765	−	0.572700i	\(-0.805896\pi\)
							−0.738000	+	0.674801i	\(0.764229\pi\)
\(31\)	4.82349	−	2.78484i	0.866324	−	0.500172i	0.000199064	−	1.00000i	\(-0.499937\pi\)
							0.866125	+	0.499828i	\(0.166603\pi\)
\(37\)	−0.768663	−	3.86433i	−0.126367	−	0.635292i	−0.991107	−	0.133070i	\(-0.957516\pi\)
							0.864739	−	0.502221i	\(-0.167484\pi\)
\(41\)	0.345101	+	2.62130i	0.0538957	+	0.409378i	0.997033	+	0.0769750i	\(0.0245262\pi\)
							−0.943137	+	0.332403i	\(0.892141\pi\)
\(43\)	11.4323	−	5.63777i	1.74340	−	0.859752i	0.767654	−	0.640864i	\(-0.221424\pi\)
							0.975750	−	0.218888i	\(-0.0702428\pi\)
\(47\)	0.949695	−	0.254470i	0.138527	−	0.0371183i	−0.188889	−	0.981998i	\(-0.560489\pi\)
							0.327416	+	0.944880i	\(0.393822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.29	yes	1504
9.4	even	3	inner	576.2.bm.a.13.93	✓	1504
64.5	even	16	inner	576.2.bm.a.133.93	yes	1504
576.517	even	48	inner	576.2.bm.a.517.29	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.93	✓	1504	9.4	even	3	inner
576.2.bm.a.133.93	yes	1504	64.5	even	16	inner
576.2.bm.a.205.29	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.29	yes	1504	576.517	even	48	inner