Embedded newform 576.2.bm.a.205.22

• Label: 576.2.bm.a.205.22
• 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			205.22
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09539 - 0.894500i) q^{2} +(-1.71899 - 0.212286i) q^{3} +(0.399740 + 1.95964i) q^{4} +(2.09926 + 2.39375i) q^{5} +(1.69307 + 1.77017i) q^{6} +(-0.401915 - 0.0529131i) q^{7} +(1.31503 - 2.50413i) q^{8} +(2.90987 + 0.729836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09539	−	0.894500i	−0.774555	−	0.632507i
							\(3\)	−1.71899	−	0.212286i	−0.992461	−	0.122563i
\(5\)	2.09926	+	2.39375i	0.938819	+	1.07052i								0.997388	+	0.0722349i	\(0.0230131\pi\)
							−0.0585683	+	0.998283i	\(0.518654\pi\)
\(7\)	−0.401915	−	0.0529131i	−0.151910	−	0.0199993i	0.0541878	−	0.998531i	\(-0.482743\pi\)
							−0.206097	+	0.978531i	\(0.566076\pi\)
\(11\)	−0.0136052	−	0.0275886i	−0.00410212	−	0.00831827i	0.894812	−	0.446443i	\(-0.147309\pi\)
							−0.898914	+	0.438125i	\(0.855643\pi\)
\(13\)	0.691828	+	2.03806i	0.191879	+	0.565256i	0.999708	−	0.0241601i	\(-0.00769114\pi\)
							−0.807829	+	0.589416i	\(0.799358\pi\)
\(17\)	−1.86727	−	1.86727i	−0.452879	−	0.452879i	0.443430	−	0.896309i	\(-0.353761\pi\)
							−0.896309	+	0.443430i	\(0.853761\pi\)
\(19\)	−0.587985	+	2.95600i	−0.134893	+	0.678153i	0.852862	+	0.522137i	\(0.174865\pi\)
							−0.987755	+	0.156016i	\(0.950135\pi\)
\(23\)	0.0505938	+	0.384298i	0.0105495	+	0.0801316i	0.995910	−	0.0903538i	\(-0.0287998\pi\)
							−0.985360	+	0.170485i	\(0.945466\pi\)
\(29\)	3.58256	−	0.234814i	0.665265	−	0.0436038i	0.270970	−	0.962588i	\(-0.412655\pi\)
							0.394295	+	0.918984i	\(0.370989\pi\)
\(31\)	−7.70724	+	4.44978i	−1.38426	+	0.799203i	−0.992661	−	0.120932i	\(-0.961412\pi\)
							−0.391600	+	0.920136i	\(0.628078\pi\)
\(37\)	1.68150	+	8.45345i	0.276436	+	1.38974i	0.830386	+	0.557189i	\(0.188120\pi\)
							−0.553950	+	0.832550i	\(0.686880\pi\)
\(41\)	0.682674	+	5.18542i	0.106616	+	0.809827i	0.958298	+	0.285769i	\(0.0922490\pi\)
							−0.851683	+	0.524058i	\(0.824418\pi\)
\(43\)	0.855136	−	0.421706i	0.130407	−	0.0643096i	−0.375915	−	0.926654i	\(-0.622671\pi\)
							0.506322	+	0.862345i	\(0.331005\pi\)
\(47\)	7.12206	−	1.90835i	1.03886	−	0.278362i	0.301219	−	0.953555i	\(-0.402607\pi\)
							0.737641	+	0.675193i	\(0.235940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.22	yes	1504
9.4	even	3	inner	576.2.bm.a.13.84	✓	1504
64.5	even	16	inner	576.2.bm.a.133.84	yes	1504
576.517	even	48	inner	576.2.bm.a.517.22	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.84	✓	1504	9.4	even	3	inner
576.2.bm.a.133.84	yes	1504	64.5	even	16	inner
576.2.bm.a.205.22	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.22	yes	1504	576.517	even	48	inner