Embedded newform 576.2.bm.a.205.81

• Label: 576.2.bm.a.205.81
• 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			205.81
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.81

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.26862 - 0.624986i) q^{2} +(0.743116 - 1.56454i) q^{3} +(1.21879 - 1.58574i) q^{4} +(-2.14098 - 2.44132i) q^{5} +(-0.0350837 - 2.44924i) q^{6} +(3.97485 + 0.523299i) q^{7} +(0.555110 - 2.77342i) q^{8} +(-1.89556 - 2.32527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.26862	−	0.624986i	0.897049	−	0.441932i
							\(3\)	0.743116	−	1.56454i	0.429038	−	0.903286i
\(5\)	−2.14098	−	2.44132i	−0.957477	−	1.09179i								−0.995710	−	0.0925241i	\(-0.970506\pi\)
							0.0382338	−	0.999269i	\(-0.487827\pi\)
\(7\)	3.97485	+	0.523299i	1.50235	+	0.197788i	0.836364	−	0.548174i	\(-0.184677\pi\)
							0.665988	+	0.745963i	\(0.268010\pi\)
\(11\)	2.21873	+	4.49914i	0.668973	+	1.35654i	0.921282	+	0.388896i	\(0.127144\pi\)
							−0.252308	+	0.967647i	\(0.581190\pi\)
\(13\)	1.12131	+	3.30328i	0.310996	+	0.916164i	0.984254	+	0.176757i	\(0.0565607\pi\)
							−0.673259	+	0.739407i	\(0.735106\pi\)
\(17\)	−3.35193	−	3.35193i	−0.812963	−	0.812963i	0.172114	−	0.985077i	\(-0.444940\pi\)
							−0.985077	+	0.172114i	\(0.944940\pi\)
\(19\)	−0.689285	+	3.46527i	−0.158133	+	0.794988i	0.817560	+	0.575844i	\(0.195326\pi\)
							−0.975693	+	0.219144i	\(0.929674\pi\)
\(23\)	0.198942	+	1.51111i	0.0414822	+	0.315089i	0.999589	+	0.0286767i	\(0.00912934\pi\)
							−0.958106	+	0.286412i	\(0.907537\pi\)
\(29\)	−1.86286	+	0.122098i	−0.345924	+	0.0226731i	−0.237375	−	0.971418i	\(-0.576287\pi\)
							−0.108549	+	0.994091i	\(0.534620\pi\)
\(31\)	−5.71856	+	3.30161i	−1.02708	+	0.592987i	−0.916148	−	0.400841i	\(-0.868718\pi\)
							−0.110935	+	0.993828i	\(0.535385\pi\)
\(37\)	0.865069	+	4.34900i	0.142216	+	0.714970i	0.984423	+	0.175816i	\(0.0562562\pi\)
							−0.842207	+	0.539155i	\(0.818744\pi\)
\(41\)	−0.763920	−	5.80255i	−0.119304	−	0.906206i	−0.941503	−	0.337005i	\(-0.890586\pi\)
							0.822199	−	0.569201i	\(-0.192747\pi\)
\(43\)	8.98256	−	4.42971i	1.36983	−	0.675524i	0.399368	−	0.916791i	\(-0.369229\pi\)
							0.970459	+	0.241266i	\(0.0775628\pi\)
\(47\)	10.5898	−	2.83753i	1.54468	−	0.413896i	0.616906	−	0.787037i	\(-0.288386\pi\)
							0.927775	+	0.373141i	\(0.121719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.81	yes	1504
9.4	even	3	inner	576.2.bm.a.13.46	✓	1504
64.5	even	16	inner	576.2.bm.a.133.46	yes	1504
576.517	even	48	inner	576.2.bm.a.517.81	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.46	✓	1504	9.4	even	3	inner
576.2.bm.a.133.46	yes	1504	64.5	even	16	inner
576.2.bm.a.205.81	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.81	yes	1504	576.517	even	48	inner