Embedded newform 576.2.bm.a.205.52

• Label: 576.2.bm.a.205.52
• 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			205.52
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.210640 + 1.39844i) q^{2} +(1.12489 - 1.31705i) q^{3} +(-1.91126 + 0.589135i) q^{4} +(-0.614322 - 0.700500i) q^{5} +(2.07876 + 1.29567i) q^{6} +(-2.61489 - 0.344257i) q^{7} +(-1.22646 - 2.54869i) q^{8} +(-0.469232 - 2.96308i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.210640	+	1.39844i	0.148945	+	0.988845i
							\(3\)	1.12489	−	1.31705i	0.649457	−	0.760398i
\(5\)	−0.614322	−	0.700500i	−0.274733	−	0.313273i								0.597976	−	0.801514i	\(-0.295972\pi\)
							−0.872709	+	0.488241i	\(0.837639\pi\)
\(7\)	−2.61489	−	0.344257i	−0.988336	−	0.130117i	−0.381011	−	0.924570i	\(-0.624424\pi\)
							−0.607325	+	0.794454i	\(0.707757\pi\)
\(11\)	−0.249773	−	0.506490i	−0.0753094	−	0.152712i	0.855962	−	0.517039i	\(-0.172966\pi\)
							−0.931271	+	0.364327i	\(0.881299\pi\)
\(13\)	−0.126180	−	0.371714i	−0.0349960	−	0.103095i	0.928028	−	0.372510i	\(-0.121503\pi\)
							−0.963024	+	0.269415i	\(0.913170\pi\)
\(17\)	−2.14512	−	2.14512i	−0.520267	−	0.520267i	0.397385	−	0.917652i	\(-0.369918\pi\)
							−0.917652	+	0.397385i	\(0.869918\pi\)
\(19\)	1.03056	−	5.18096i	0.236426	−	1.18859i	−0.662012	−	0.749493i	\(-0.730297\pi\)
							0.898438	−	0.439100i	\(-0.144703\pi\)
\(23\)	0.0230979	+	0.175446i	0.00481624	+	0.0365830i	0.993684	−	0.112212i	\(-0.0357937\pi\)
							−0.988868	+	0.148795i	\(0.952460\pi\)
\(29\)	3.44929	−	0.226078i	0.640516	−	0.0419817i	0.258315	−	0.966061i	\(-0.416833\pi\)
							0.382201	+	0.924079i	\(0.375166\pi\)
\(31\)	−6.55413	+	3.78403i	−1.17716	+	0.679631i	−0.955355	−	0.295460i	\(-0.904527\pi\)
							−0.221801	+	0.975092i	\(0.571194\pi\)
\(37\)	−0.453900	−	2.28191i	−0.0746207	−	0.375144i	0.925372	−	0.379061i	\(-0.123753\pi\)
							−0.999992	+	0.00391755i	\(0.998753\pi\)
\(41\)	−0.149016	−	1.13189i	−0.0232723	−	0.176771i	0.975861	−	0.218392i	\(-0.0700811\pi\)
							−0.999133	+	0.0416211i	\(0.986748\pi\)
\(43\)	−4.73832	+	2.33668i	−0.722587	+	0.356340i	−0.766108	−	0.642712i	\(-0.777809\pi\)
							0.0435210	+	0.999053i	\(0.486142\pi\)
\(47\)	5.24019	−	1.40411i	0.764361	−	0.204810i	0.144482	−	0.989507i	\(-0.453848\pi\)
							0.619879	+	0.784698i	\(0.287182\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.52	yes	1504
9.4	even	3	inner	576.2.bm.a.13.12	✓	1504
64.5	even	16	inner	576.2.bm.a.133.12	yes	1504
576.517	even	48	inner	576.2.bm.a.517.52	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.12	✓	1504	9.4	even	3	inner
576.2.bm.a.133.12	yes	1504	64.5	even	16	inner
576.2.bm.a.205.52	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.52	yes	1504	576.517	even	48	inner