Embedded newform 576.2.bm.a.205.60

• Label: 576.2.bm.a.205.60
• 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			205.60
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.60

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.573780 + 1.29259i) q^{2} +(-1.48669 + 0.888688i) q^{3} +(-1.34155 + 1.48332i) q^{4} +(-1.29438 - 1.47596i) q^{5} +(-2.00174 - 1.41176i) q^{6} +(-0.101825 - 0.0134055i) q^{7} +(-2.68707 - 0.882971i) q^{8} +(1.42047 - 2.64240i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.573780	+	1.29259i	0.405724	+	0.913996i
							\(3\)	−1.48669	+	0.888688i	−0.858338	+	0.513084i
\(5\)	−1.29438	−	1.47596i	−0.578865	−	0.660069i								0.386097	−	0.922458i	\(-0.373823\pi\)
							−0.964962	+	0.262389i	\(0.915490\pi\)
\(7\)	−0.101825	−	0.0134055i	−0.0384862	−	0.00506680i	0.111259	−	0.993791i	\(-0.464512\pi\)
							−0.149745	+	0.988725i	\(0.547845\pi\)
\(11\)	−1.90532	−	3.86361i	−0.574477	−	1.16492i	−0.969235	−	0.246138i	\(-0.920838\pi\)
							0.394758	−	0.918785i	\(-0.370828\pi\)
\(13\)	1.10863	+	3.26592i	0.307478	+	0.905802i	0.985338	+	0.170611i	\(0.0545742\pi\)
							−0.677860	+	0.735191i	\(0.737092\pi\)
\(17\)	4.05406	+	4.05406i	0.983254	+	0.983254i	0.999862	−	0.0166082i	\(-0.00528679\pi\)
							−0.0166082	+	0.999862i	\(0.505287\pi\)
\(19\)	1.11476	−	5.60428i	0.255744	−	1.28571i	−0.612856	−	0.790195i	\(-0.709979\pi\)
							0.868599	−	0.495515i	\(-0.165021\pi\)
\(23\)	−1.08070	−	8.20870i	−0.225341	−	1.71163i	−0.613212	−	0.789918i	\(-0.710123\pi\)
							0.387872	−	0.921713i	\(-0.373210\pi\)
\(29\)	−5.22605	+	0.342533i	−0.970453	+	0.0636068i	−0.542376	−	0.840136i	\(-0.682475\pi\)
							−0.428076	+	0.903743i	\(0.640809\pi\)
\(31\)	1.41411	−	0.816435i	0.253981	−	0.146636i	−0.367605	−	0.929982i	\(-0.619822\pi\)
							0.621586	+	0.783346i	\(0.286489\pi\)
\(37\)	1.60924	+	8.09020i	0.264558	+	1.33002i	0.853182	+	0.521614i	\(0.174670\pi\)
							−0.588624	+	0.808407i	\(0.700330\pi\)
\(41\)	−0.445576	−	3.38449i	−0.0695873	−	0.528568i	−0.990365	−	0.138481i	\(-0.955778\pi\)
							0.920778	−	0.390087i	\(-0.127555\pi\)
\(43\)	−2.82882	+	1.39502i	−0.431391	+	0.212738i	−0.645003	−	0.764180i	\(-0.723144\pi\)
							0.213612	+	0.976919i	\(0.431477\pi\)
\(47\)	6.79974	−	1.82199i	0.991845	−	0.265764i	0.273819	−	0.961781i	\(-0.411713\pi\)
							0.718025	+	0.696017i	\(0.245046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.60	yes	1504
9.4	even	3	inner	576.2.bm.a.13.5	✓	1504
64.5	even	16	inner	576.2.bm.a.133.5	yes	1504
576.517	even	48	inner	576.2.bm.a.517.60	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.5	✓	1504	9.4	even	3	inner
576.2.bm.a.133.5	yes	1504	64.5	even	16	inner
576.2.bm.a.205.60	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.60	yes	1504	576.517	even	48	inner