Embedded newform 576.2.bm.a.517.29

• Label: 576.2.bm.a.517.29
• 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			517.29
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.29

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.93	✓	1504	64.13	even	16	inner
576.2.bm.a.133.93	yes	1504	9.7	even	3	inner
576.2.bm.a.205.29	yes	1504	576.205	even	48	inner
576.2.bm.a.517.29	yes	1504	1.1	even	1	trivial