Embedded newform 576.2.bm.a.205.31

• Label: 576.2.bm.a.205.31
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.31
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.775373 - 1.18271i) q^{2} +(0.720990 + 1.57486i) q^{3} +(-0.797593 + 1.83408i) q^{4} +(-0.992718 - 1.13198i) q^{5} +(1.30356 - 2.07382i) q^{6} +(-1.51707 - 0.199726i) q^{7} +(2.78761 - 0.478776i) q^{8} +(-1.96035 + 2.27091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.775373	−	1.18271i	−0.548272	−	0.836300i
							\(3\)	0.720990	+	1.57486i	0.416263	+	0.909244i
\(5\)	−0.992718	−	1.13198i	−0.443957	−	0.506236i								0.485972	−	0.873974i	\(-0.338466\pi\)
							−0.929929	+	0.367738i	\(0.880132\pi\)
\(7\)	−1.51707	−	0.199726i	−0.573397	−	0.0754892i	−0.161749	−	0.986832i	\(-0.551713\pi\)
							−0.411648	+	0.911343i	\(0.635047\pi\)
\(11\)	−1.85338	−	3.75828i	−0.558815	−	1.13317i	−0.974652	−	0.223726i	\(-0.928178\pi\)
							0.415837	−	0.909439i	\(-0.363489\pi\)
\(13\)	0.737900	+	2.17378i	0.204657	+	0.602899i	0.999995	−	0.00326703i	\(-0.00103993\pi\)
							−0.795338	+	0.606166i	\(0.792707\pi\)
\(17\)	−4.20334	−	4.20334i	−1.01946	−	1.01946i	−0.999807	−	0.0196524i	\(-0.993744\pi\)
							−0.0196524	−	0.999807i	\(-0.506256\pi\)
\(19\)	0.0676413	−	0.340056i	0.0155180	−	0.0780141i	−0.972250	−	0.233943i	\(-0.924837\pi\)
							0.987768	+	0.155929i	\(0.0498371\pi\)
\(23\)	0.115392	+	0.876493i	0.0240610	+	0.182761i	0.999255	−	0.0385899i	\(-0.0122866\pi\)
							−0.975194	+	0.221351i	\(0.928953\pi\)
\(29\)	1.53182	−	0.100401i	0.284453	−	0.0186440i	0.0774909	−	0.996993i	\(-0.475309\pi\)
							0.206962	+	0.978349i	\(0.433642\pi\)
\(31\)	−3.11702	+	1.79961i	−0.559834	+	0.323220i	−0.753079	−	0.657930i	\(-0.771432\pi\)
							0.193245	+	0.981151i	\(0.438099\pi\)
\(37\)	−1.96962	−	9.90194i	−0.323803	−	1.62787i	−0.709122	−	0.705086i	\(-0.750908\pi\)
							0.385318	−	0.922784i	\(-0.374092\pi\)
\(41\)	−1.45873	−	11.0801i	−0.227815	−	1.73042i	−0.598026	−	0.801477i	\(-0.704048\pi\)
							0.370211	−	0.928948i	\(-0.379285\pi\)
\(43\)	−6.89969	+	3.40255i	−1.05219	+	0.518884i	−0.884299	−	0.466921i	\(-0.845363\pi\)
							−0.167894	+	0.985805i	\(0.553697\pi\)
\(47\)	5.41876	−	1.45195i	0.790408	−	0.211789i	0.159039	−	0.987272i	\(-0.449160\pi\)
							0.631368	+	0.775483i	\(0.282494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.31	yes	1504
9.4	even	3	inner	576.2.bm.a.13.94	✓	1504
64.5	even	16	inner	576.2.bm.a.133.94	yes	1504
576.517	even	48	inner	576.2.bm.a.517.31	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.94	✓	1504	9.4	even	3	inner
576.2.bm.a.133.94	yes	1504	64.5	even	16	inner
576.2.bm.a.205.31	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.31	yes	1504	576.517	even	48	inner