Embedded newform 576.2.bl.a.299.57

• Label: 576.2.bl.a.299.57
• Level: $576$
• Weight: $2$
• Character: 576.299
• 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.bl (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			299.57
Character	\(\chi\)	\(=\)	576.299
Dual form			576.2.bl.a.131.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.488504 + 1.32716i) q^{2} +(0.847205 - 1.51071i) q^{3} +(-1.52273 + 1.29665i) q^{4} +(0.514607 - 1.04352i) q^{5} +(2.41882 + 0.386392i) q^{6} +(0.187771 + 0.244708i) q^{7} +(-2.46472 - 1.38749i) q^{8} +(-1.56449 - 2.55976i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.488504	+	1.32716i	0.345424	+	0.938447i
							\(3\)	0.847205	−	1.51071i	0.489134	−	0.872209i
\(5\)	0.514607	−	1.04352i	0.230139	−	0.466677i								−0.750918	−	0.660396i	\(-0.770388\pi\)
							0.981057	+	0.193719i	\(0.0620551\pi\)
\(7\)	0.187771	+	0.244708i	0.0709707	+	0.0924908i	0.827490	−	0.561481i	\(-0.189768\pi\)
							−0.756519	+	0.653972i	\(0.773102\pi\)
\(11\)	3.29021	−	3.75177i	0.992036	−	1.13120i	0.000680077	−	1.00000i	\(-0.499784\pi\)
							0.991356	−	0.131200i	\(-0.0418831\pi\)
\(13\)	0.158010	−	2.41077i	0.0438241	−	0.668626i	−0.918278	−	0.395936i	\(-0.870420\pi\)
							0.962102	−	0.272690i	\(-0.0879132\pi\)
\(17\)	2.55365	+	2.55365i	0.619351	+	0.619351i	0.945365	−	0.326014i	\(-0.105706\pi\)
							−0.326014	+	0.945365i	\(0.605706\pi\)
\(19\)	−1.60114	−	1.06984i	−0.367326	−	0.245439i	0.358184	−	0.933651i	\(-0.383396\pi\)
							−0.725510	+	0.688212i	\(0.758396\pi\)
\(23\)	1.13072	−	1.47359i	0.235772	−	0.307264i	−0.660450	−	0.750870i	\(-0.729634\pi\)
							0.896222	+	0.443606i	\(0.146301\pi\)
\(29\)	−3.26084	−	1.10691i	−0.605523	−	0.205547i	0.00180546	−	0.999998i	\(-0.499425\pi\)
							−0.607328	+	0.794451i	\(0.707759\pi\)
\(31\)	3.19670	+	5.53685i	0.574145	+	0.994448i	0.996134	+	0.0878469i	\(0.0279986\pi\)
							−0.421989	+	0.906601i	\(0.638668\pi\)
\(37\)	−4.02189	−	6.01919i	−0.661195	−	0.989548i	−0.998834	−	0.0482728i	\(-0.984628\pi\)
							0.337639	−	0.941276i	\(-0.390372\pi\)
\(41\)	3.33245	+	2.55708i	0.520441	+	0.399348i	0.835304	−	0.549789i	\(-0.185292\pi\)
							−0.314863	+	0.949137i	\(0.601959\pi\)
\(43\)	6.49409	−	7.40510i	0.990340	−	1.12927i	−0.00126964	−	0.999999i	\(-0.500404\pi\)
							0.991610	−	0.129267i	\(-0.0412625\pi\)
\(47\)	2.20023	+	8.21139i	0.320937	+	1.19775i	0.918333	+	0.395808i	\(0.129535\pi\)
							−0.597396	+	0.801946i	\(0.703798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.299.57	yes	1504
9.5	odd	6	inner	576.2.bl.a.491.90	yes	1504
64.3	odd	16	inner	576.2.bl.a.515.90	yes	1504
576.131	even	48	inner	576.2.bl.a.131.57	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.131.57	✓	1504	576.131	even	48	inner
576.2.bl.a.299.57	yes	1504	1.1	even	1	trivial
576.2.bl.a.491.90	yes	1504	9.5	odd	6	inner
576.2.bl.a.515.90	yes	1504	64.3	odd	16	inner