Embedded newform 576.2.bm.a.13.17

• Label: 576.2.bm.a.13.17
• Level: $576$
• Weight: $2$
• Character: 576.13
• 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			13.17
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18741 + 0.768145i) q^{2} +(0.146556 + 1.72584i) q^{3} +(0.819907 - 1.82421i) q^{4} +(-0.0104531 + 0.0307939i) q^{5} +(-1.49972 - 1.93671i) q^{6} +(1.55545 - 2.02711i) q^{7} +(0.427690 + 2.79590i) q^{8} +(-2.95704 + 0.505864i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18741	+	0.768145i	−0.839629	+	0.543160i
							\(3\)	0.146556	+	1.72584i	0.0846142	+	0.996414i
\(5\)	−0.0104531	+	0.0307939i	−0.00467478	+	0.0137715i								−0.949242	−	0.314545i	\(-0.898148\pi\)
							0.944568	+	0.328317i	\(0.106481\pi\)
\(7\)	1.55545	−	2.02711i	0.587907	−	0.766175i	−0.400779	−	0.916175i	\(-0.631261\pi\)
							0.988686	+	0.150000i	\(0.0479273\pi\)
\(11\)	0.378604	+	0.0248150i	0.114153	+	0.00748200i	0.122373	−	0.992484i	\(-0.460950\pi\)
							−0.00821941	+	0.999966i	\(0.502616\pi\)
\(13\)	3.94967	−	4.50373i	1.09544	−	1.24911i	0.129159	−	0.991624i	\(-0.458772\pi\)
							0.966282	−	0.257487i	\(-0.0828946\pi\)
\(17\)	3.22106	+	3.22106i	0.781221	+	0.781221i	0.980037	−	0.198816i	\(-0.0637096\pi\)
							−0.198816	+	0.980037i	\(0.563710\pi\)
\(19\)	−0.645361	+	3.24445i	−0.148056	+	0.744328i	0.833405	+	0.552663i	\(0.186388\pi\)
							−0.981461	+	0.191664i	\(0.938612\pi\)
\(23\)	3.14455	−	2.41290i	0.655684	−	0.503124i	−0.226511	−	0.974009i	\(-0.572732\pi\)
							0.882195	+	0.470885i	\(0.156065\pi\)
\(29\)	1.51390	−	3.06988i	0.281123	−	0.570062i	−0.709814	−	0.704390i	\(-0.751221\pi\)
							0.990937	+	0.134328i	\(0.0428875\pi\)
\(31\)	0.940355	+	0.542914i	0.168893	+	0.0975103i	0.582064	−	0.813143i	\(-0.302245\pi\)
							−0.413171	+	0.910654i	\(0.635579\pi\)
\(37\)	−0.646259	−	3.24896i	−0.106244	−	0.534126i	−0.996847	−	0.0793444i	\(-0.974717\pi\)
							0.890603	−	0.454782i	\(-0.150283\pi\)
\(41\)	2.27797	−	1.74795i	0.355760	−	0.272984i	−0.415383	−	0.909647i	\(-0.636352\pi\)
							0.771142	+	0.636663i	\(0.219686\pi\)
\(43\)	−0.0908339	+	1.38586i	−0.0138520	+	0.211341i	0.985322	+	0.170706i	\(0.0546048\pi\)
							−0.999174	+	0.0406353i	\(0.987062\pi\)
\(47\)	−2.80813	+	10.4801i	−0.409608	+	1.52868i	0.385787	+	0.922588i	\(0.373930\pi\)
							−0.795395	+	0.606091i	\(0.792737\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.17	✓	1504
9.7	even	3	inner	576.2.bm.a.205.79	yes	1504
64.5	even	16	inner	576.2.bm.a.517.79	yes	1504
576.133	even	48	inner	576.2.bm.a.133.17	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.17	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.17	yes	1504	576.133	even	48	inner
576.2.bm.a.205.79	yes	1504	9.7	even	3	inner
576.2.bm.a.517.79	yes	1504	64.5	even	16	inner