Embedded newform 576.2.bm.a.517.54

• Label: 576.2.bm.a.517.54
• 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			517.54
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.325071 - 1.37635i) q^{2} +(-1.01717 + 1.40191i) q^{3} +(-1.78866 - 0.894820i) q^{4} +(-2.77927 + 3.16915i) q^{5} +(1.59887 + 1.85570i) q^{6} +(-1.04166 + 0.137138i) q^{7} +(-1.81302 + 2.17093i) q^{8} +(-0.930729 - 2.85197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.325071	−	1.37635i	0.229860	−	0.973224i
							\(3\)	−1.01717	+	1.40191i	−0.587264	+	0.809396i
\(5\)	−2.77927	+	3.16915i	−1.24293	+	1.41729i								−0.370294	+	0.928915i	\(0.620743\pi\)
							−0.872635	+	0.488374i	\(0.837590\pi\)
\(7\)	−1.04166	+	0.137138i	−0.393712	+	0.0518332i	−0.324783	−	0.945788i	\(-0.605291\pi\)
							−0.0689286	+	0.997622i	\(0.521958\pi\)
\(11\)	2.72423	−	5.52419i	0.821385	−	1.66560i	0.0777948	−	0.996969i	\(-0.475212\pi\)
							0.743591	−	0.668635i	\(-0.233121\pi\)
\(13\)	−0.213037	+	0.627587i	−0.0590859	+	0.174061i	−0.972465	−	0.233048i	\(-0.925130\pi\)
							0.913379	+	0.407110i	\(0.133463\pi\)
\(17\)	3.81355	−	3.81355i	0.924921	−	0.924921i	−0.0724509	−	0.997372i	\(-0.523082\pi\)
							0.997372	+	0.0724509i	\(0.0230820\pi\)
\(19\)	0.464562	+	2.33551i	0.106578	+	0.535803i	0.996776	+	0.0802300i	\(0.0255655\pi\)
							−0.890199	+	0.455573i	\(0.849435\pi\)
\(23\)	0.493898	−	3.75153i	0.102985	−	0.782247i	−0.859543	−	0.511064i	\(-0.829252\pi\)
							0.962528	−	0.271183i	\(-0.0874150\pi\)
\(29\)	−4.78009	−	0.313304i	−0.887641	−	0.0581790i	−0.385242	−	0.922815i	\(-0.625882\pi\)
							−0.502398	+	0.864636i	\(0.667549\pi\)
\(31\)	1.86974	+	1.07949i	0.335815	+	0.193883i	0.658420	−	0.752651i	\(-0.271225\pi\)
							−0.322605	+	0.946534i	\(0.604559\pi\)
\(37\)	0.122483	−	0.615761i	0.0201360	−	0.101231i	−0.969410	−	0.245446i	\(-0.921066\pi\)
							0.989546	+	0.144215i	\(0.0460657\pi\)
\(41\)	0.296946	−	2.25553i	0.0463752	−	0.352254i	−0.952488	−	0.304577i	\(-0.901485\pi\)
							0.998863	−	0.0476771i	\(-0.0151818\pi\)
\(43\)	8.00495	+	3.94760i	1.22074	+	0.602004i	0.934435	−	0.356134i	\(-0.115905\pi\)
							0.286308	+	0.958138i	\(0.407572\pi\)
\(47\)	−6.55930	−	1.75756i	−0.956772	−	0.256366i	−0.253539	−	0.967325i	\(-0.581594\pi\)
							−0.703233	+	0.710959i	\(0.748261\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.54	yes	1504
9.7	even	3	inner	576.2.bm.a.133.8	yes	1504
64.13	even	16	inner	576.2.bm.a.13.8	✓	1504
576.205	even	48	inner	576.2.bm.a.205.54	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.8	✓	1504	64.13	even	16	inner
576.2.bm.a.133.8	yes	1504	9.7	even	3	inner
576.2.bm.a.205.54	yes	1504	576.205	even	48	inner
576.2.bm.a.517.54	yes	1504	1.1	even	1	trivial