Embedded newform 576.2.bm.a.517.67

• Label: 576.2.bm.a.517.67
• 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			517.67
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.67

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.834872 + 1.14149i) q^{2} +(1.36695 - 1.06370i) q^{3} +(-0.605976 + 1.90599i) q^{4} +(-2.87207 + 3.27497i) q^{5} +(2.35542 + 0.672302i) q^{6} +(-1.46065 + 0.192298i) q^{7} +(-2.68157 + 0.899544i) q^{8} +(0.737100 - 2.90804i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.834872	+	1.14149i	0.590344	+	0.807152i
							\(3\)	1.36695	−	1.06370i	0.789208	−	0.614125i
\(5\)	−2.87207	+	3.27497i	−1.28443	+	1.46461i								−0.471948	+	0.881627i	\(0.656449\pi\)
							−0.812483	+	0.582985i	\(0.801884\pi\)
\(7\)	−1.46065	+	0.192298i	−0.552073	+	0.0726818i	−0.401402	−	0.915902i	\(-0.631477\pi\)
							−0.150671	+	0.988584i	\(0.548143\pi\)
\(11\)	0.246333	−	0.499513i	0.0742721	−	0.150609i	−0.856570	−	0.516031i	\(-0.827409\pi\)
							0.930842	+	0.365422i	\(0.119075\pi\)
\(13\)	−0.289120	+	0.851719i	−0.0801873	+	0.236224i	−0.979629	−	0.200817i	\(-0.935640\pi\)
							0.899441	+	0.437041i	\(0.143974\pi\)
\(17\)	−3.35029	+	3.35029i	−0.812566	+	0.812566i	−0.985018	−	0.172452i	\(-0.944831\pi\)
							0.172452	+	0.985018i	\(0.444831\pi\)
\(19\)	0.884793	+	4.44815i	0.202985	+	1.02048i	0.939106	+	0.343626i	\(0.111655\pi\)
							−0.736121	+	0.676850i	\(0.763345\pi\)
\(23\)	−0.887006	+	6.73748i	−0.184954	+	1.40486i	0.607205	+	0.794546i	\(0.292291\pi\)
							−0.792158	+	0.610316i	\(0.791042\pi\)
\(29\)	7.68288	+	0.503562i	1.42667	+	0.0935092i	0.759071	−	0.651008i	\(-0.225654\pi\)
							0.667603	+	0.744517i	\(0.267320\pi\)
\(31\)	3.00431	+	1.73454i	0.539590	+	0.311533i	0.744913	−	0.667162i	\(-0.232491\pi\)
							−0.205322	+	0.978694i	\(0.565824\pi\)
\(37\)	1.35853	−	6.82981i	0.223341	−	1.12281i	−0.692545	−	0.721375i	\(-0.743510\pi\)
							0.915886	−	0.401438i	\(-0.131490\pi\)
\(41\)	−0.158992	+	1.20767i	−0.0248304	+	0.188606i	−0.999365	−	0.0356314i	\(-0.988656\pi\)
							0.974535	+	0.224237i	\(0.0719891\pi\)
\(43\)	3.79795	+	1.87294i	0.579182	+	0.285621i	0.708189	−	0.706023i	\(-0.249512\pi\)
							−0.129007	+	0.991644i	\(0.541179\pi\)
\(47\)	1.24697	+	0.334124i	0.181889	+	0.0487370i	0.348614	−	0.937266i	\(-0.386652\pi\)
							−0.166725	+	0.986003i	\(0.553319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.67	yes	1504
9.7	even	3	inner	576.2.bm.a.133.61	yes	1504
64.13	even	16	inner	576.2.bm.a.13.61	✓	1504
576.205	even	48	inner	576.2.bm.a.205.67	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.61	✓	1504	64.13	even	16	inner
576.2.bm.a.133.61	yes	1504	9.7	even	3	inner
576.2.bm.a.205.67	yes	1504	576.205	even	48	inner
576.2.bm.a.517.67	yes	1504	1.1	even	1	trivial