Embedded newform 576.2.bm.a.517.31

• Label: 576.2.bm.a.517.31
• 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			517.31
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.31

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.94	✓	1504	64.13	even	16	inner
576.2.bm.a.133.94	yes	1504	9.7	even	3	inner
576.2.bm.a.205.31	yes	1504	576.205	even	48	inner
576.2.bm.a.517.31	yes	1504	1.1	even	1	trivial