Embedded newform 576.2.bm.a.205.39

• Label: 576.2.bm.a.205.39
• 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			205.39
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.427576 + 1.34803i) q^{2} +(1.64520 - 0.541597i) q^{3} +(-1.63436 - 1.15277i) q^{4} +(2.38016 + 2.71405i) q^{5} +(0.0266422 + 2.44934i) q^{6} +(3.65079 + 0.480636i) q^{7} +(2.25278 - 1.71026i) q^{8} +(2.41334 - 1.78207i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.427576	+	1.34803i	−0.302342	+	0.953200i
							\(3\)	1.64520	−	0.541597i	0.949855	−	0.312691i
\(5\)	2.38016	+	2.71405i	1.06444	+	1.21376i								0.976020	+	0.217679i	\(0.0698486\pi\)
							0.0884205	+	0.996083i	\(0.471818\pi\)
\(7\)	3.65079	+	0.480636i	1.37987	+	0.181663i	0.783672	−	0.621175i	\(-0.213344\pi\)
							0.596197	+	0.802838i	\(0.296678\pi\)
\(11\)	−1.82388	−	3.69846i	−0.549920	−	1.11513i	−0.977458	−	0.211130i	\(-0.932286\pi\)
							0.427538	−	0.903998i	\(-0.359381\pi\)
\(13\)	1.01627	+	2.99385i	0.281864	+	0.830344i	0.991995	+	0.126275i	\(0.0403021\pi\)
							−0.710132	+	0.704069i	\(0.751365\pi\)
\(17\)	−4.41542	−	4.41542i	−1.07090	−	1.07090i	−0.997287	−	0.0736098i	\(-0.976548\pi\)
							−0.0736098	−	0.997287i	\(-0.523452\pi\)
\(19\)	0.330635	−	1.66221i	0.0758529	−	0.381338i	−0.924147	−	0.382037i	\(-0.875223\pi\)
							1.00000		0.000699247i	\(0.000222577\pi\)
\(23\)	−0.580807	−	4.41167i	−0.121107	−	0.919896i	−0.938853	−	0.344318i	\(-0.888110\pi\)
							0.817747	−	0.575578i	\(-0.195223\pi\)
\(29\)	−4.75151	+	0.311431i	−0.882334	+	0.0578312i	−0.499831	−	0.866123i	\(-0.666605\pi\)
							−0.382503	+	0.923954i	\(0.624938\pi\)
\(31\)	−4.26195	+	2.46064i	−0.765469	+	0.441944i	−0.831256	−	0.555890i	\(-0.812378\pi\)
							0.0657867	+	0.997834i	\(0.479044\pi\)
\(37\)	1.25696	+	6.31914i	0.206642	+	1.03886i	0.935266	+	0.353946i	\(0.115160\pi\)
							−0.728624	+	0.684914i	\(0.759840\pi\)
\(41\)	−0.274218	−	2.08289i	−0.0428256	−	0.325293i	−0.999426	−	0.0338885i	\(-0.989211\pi\)
							0.956600	−	0.291404i	\(-0.0941224\pi\)
\(43\)	−5.84251	+	2.88121i	−0.890974	+	0.439380i	−0.829356	−	0.558720i	\(-0.811293\pi\)
							−0.0616178	+	0.998100i	\(0.519626\pi\)
\(47\)	0.735963	−	0.197201i	0.107351	−	0.0287647i	−0.204743	−	0.978816i	\(-0.565636\pi\)
							0.312095	+	0.950051i	\(0.398969\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.39	yes	1504
9.4	even	3	inner	576.2.bm.a.13.24	✓	1504
64.5	even	16	inner	576.2.bm.a.133.24	yes	1504
576.517	even	48	inner	576.2.bm.a.517.39	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.24	✓	1504	9.4	even	3	inner
576.2.bm.a.133.24	yes	1504	64.5	even	16	inner
576.2.bm.a.205.39	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.39	yes	1504	576.517	even	48	inner