Embedded newform 576.2.bl.a.155.77

• Label: 576.2.bl.a.155.77
• Level: $576$
• Weight: $2$
• Character: 576.155
• 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			155.77
Character	\(\chi\)	\(=\)	576.155
Dual form			576.2.bl.a.275.77

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18808 + 0.767120i) q^{2} +(-1.59080 + 0.685103i) q^{3} +(0.823055 + 1.82279i) q^{4} +(0.821400 + 2.41977i) q^{5} +(-2.41555 - 0.406375i) q^{6} +(-2.87209 + 2.20383i) q^{7} +(-0.420448 + 2.79700i) q^{8} +(2.06127 - 2.17972i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18808	+	0.767120i	0.840097	+	0.542435i
							\(3\)	−1.59080	+	0.685103i	−0.918447	+	0.395545i
\(5\)	0.821400	+	2.41977i	0.367341	+	1.08215i								0.960789	+	0.277280i	\(0.0894331\pi\)
							−0.593448	+	0.804873i	\(0.702234\pi\)
\(7\)	−2.87209	+	2.20383i	−1.08555	+	0.832970i	−0.986768	−	0.162139i	\(-0.948161\pi\)
							−0.0987800	+	0.995109i	\(0.531494\pi\)
\(11\)	0.0111093	+	0.169495i	0.00334957	+	0.0511046i	0.999206	−	0.0398294i	\(-0.0126814\pi\)
							−0.995857	+	0.0909340i	\(0.971015\pi\)
\(13\)	2.52388	−	2.21339i	0.699999	−	0.613883i	−0.233545	−	0.972346i	\(-0.575032\pi\)
							0.933544	+	0.358463i	\(0.116699\pi\)
\(17\)	−2.69966	−	2.69966i	−0.654764	−	0.654764i	0.299372	−	0.954136i	\(-0.403223\pi\)
							−0.954136	+	0.299372i	\(0.903223\pi\)
\(19\)	−0.102664	−	0.516128i	−0.0235528	−	0.118408i	0.967220	−	0.253938i	\(-0.0817261\pi\)
							−0.990773	+	0.135531i	\(0.956726\pi\)
\(23\)	−3.32766	−	2.55340i	−0.693865	−	0.532421i	0.200537	−	0.979686i	\(-0.435731\pi\)
							−0.894402	+	0.447265i	\(0.852398\pi\)
\(29\)	2.29000	+	4.64366i	0.425242	+	0.862305i	0.999061	+	0.0433234i	\(0.0137946\pi\)
							−0.573819	+	0.818982i	\(0.694539\pi\)
\(31\)	4.64410	+	8.04382i	0.834106	+	1.44471i	0.894756	+	0.446554i	\(0.147349\pi\)
							−0.0606507	+	0.998159i	\(0.519318\pi\)
\(37\)	−2.76349	−	0.549693i	−0.454315	−	0.0903689i	−0.0373720	−	0.999301i	\(-0.511899\pi\)
							−0.416943	+	0.908932i	\(0.636899\pi\)
\(41\)	−1.55355	+	2.02463i	−0.242624	+	0.316194i	−0.898753	−	0.438455i	\(-0.855526\pi\)
							0.656129	+	0.754649i	\(0.272193\pi\)
\(43\)	−0.0771319	−	1.17680i	−0.0117625	−	0.179461i	−0.999698	−	0.0245924i	\(-0.992171\pi\)
							0.987935	−	0.154869i	\(-0.0494955\pi\)
\(47\)	2.09832	+	7.83103i	0.306071	+	1.14227i	0.932019	+	0.362408i	\(0.118045\pi\)
							−0.625948	+	0.779865i	\(0.715288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.155.77	yes	1504
9.5	odd	6	inner	576.2.bl.a.347.82	yes	1504
64.19	odd	16	inner	576.2.bl.a.83.82	✓	1504
576.275	even	48	inner	576.2.bl.a.275.77	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.82	✓	1504	64.19	odd	16	inner
576.2.bl.a.155.77	yes	1504	1.1	even	1	trivial
576.2.bl.a.275.77	yes	1504	576.275	even	48	inner
576.2.bl.a.347.82	yes	1504	9.5	odd	6	inner