Embedded newform 576.2.bl.a.347.41

• Label: 576.2.bl.a.347.41
• Level: $576$
• Weight: $2$
• Character: 576.347
• 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.bl (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			347.41
Character	\(\chi\)	\(=\)	576.347
Dual form			576.2.bl.a.83.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326341 + 1.37605i) q^{2} +(-1.10447 + 1.33423i) q^{3} +(-1.78700 - 0.898119i) q^{4} +(0.390464 - 0.445238i) q^{5} +(-1.47552 - 1.95521i) q^{6} +(-0.227235 - 1.72602i) q^{7} +(1.81903 - 2.16591i) q^{8} +(-0.560313 - 2.94721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.326341	+	1.37605i	−0.230758	+	0.973011i
							\(3\)	−1.10447	+	1.33423i	−0.637663	+	0.770315i
\(5\)	0.390464	−	0.445238i	0.174621	−	0.199117i								−0.657909	−	0.753098i	\(-0.728559\pi\)
							0.832529	+	0.553981i	\(0.186892\pi\)
\(7\)	−0.227235	−	1.72602i	−0.0858866	−	0.652373i	−0.979194	−	0.202927i	\(-0.934955\pi\)
							0.893307	−	0.449446i	\(-0.148379\pi\)
\(11\)	−2.31157	−	1.13994i	−0.696965	−	0.343705i	0.0590540	−	0.998255i	\(-0.481192\pi\)
							−0.756019	+	0.654550i	\(0.772858\pi\)
\(13\)	6.27310	+	2.12943i	1.73984	+	0.590597i	0.996894	−	0.0787502i	\(-0.0250929\pi\)
							0.742949	+	0.669348i	\(0.233426\pi\)
\(17\)	3.24939	+	3.24939i	0.788093	+	0.788093i	0.981181	−	0.193088i	\(-0.0618503\pi\)
							−0.193088	+	0.981181i	\(0.561850\pi\)
\(19\)	0.0330210	+	0.166008i	0.00757555	+	0.0380848i	0.984388	−	0.176011i	\(-0.0563195\pi\)
							−0.976813	+	0.214096i	\(0.931319\pi\)
\(23\)	0.930230	−	7.06580i	0.193966	−	1.47332i	−0.566563	−	0.824018i	\(-0.691727\pi\)
							0.760530	−	0.649303i	\(-0.224939\pi\)
\(29\)	−2.87033	−	0.188132i	−0.533008	−	0.0349352i	−0.203476	−	0.979080i	\(-0.565224\pi\)
							−0.329531	+	0.944145i	\(0.606891\pi\)
\(31\)	3.97508	−	6.88504i	0.713946	−	1.23659i	−0.249419	−	0.968396i	\(-0.580240\pi\)
							0.963365	−	0.268194i	\(-0.0864270\pi\)
\(37\)	−1.20923	−	0.240530i	−0.198796	−	0.0395429i	0.0946880	−	0.995507i	\(-0.469815\pi\)
							−0.293484	+	0.955964i	\(0.594815\pi\)
\(41\)	−6.23888	−	0.821364i	−0.974349	−	0.128276i	−0.373493	−	0.927633i	\(-0.621840\pi\)
							−0.600856	+	0.799357i	\(0.705173\pi\)
\(43\)	4.55188	+	2.24474i	0.694156	+	0.342320i	0.754906	−	0.655833i	\(-0.227683\pi\)
							−0.0607501	+	0.998153i	\(0.519349\pi\)
\(47\)	6.05784	+	1.62319i	0.883627	+	0.236767i	0.671971	−	0.740577i	\(-0.265448\pi\)
							0.211656	+	0.977344i	\(0.432114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.347.41	yes	1504
9.2	odd	6	inner	576.2.bl.a.155.9	yes	1504
64.19	odd	16	inner	576.2.bl.a.275.9	yes	1504
576.83	even	48	inner	576.2.bl.a.83.41	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.41	✓	1504	576.83	even	48	inner
576.2.bl.a.155.9	yes	1504	9.2	odd	6	inner
576.2.bl.a.275.9	yes	1504	64.19	odd	16	inner
576.2.bl.a.347.41	yes	1504	1.1	even	1	trivial