Embedded newform 576.2.bl.a.11.14

• Label: 576.2.bl.a.11.14
• Level: $576$
• Weight: $2$
• Character: 576.11
• 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			11.14
Character	\(\chi\)	\(=\)	576.11
Dual form			576.2.bl.a.419.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26628 + 0.629706i) q^{2} +(1.72410 + 0.165818i) q^{3} +(1.20694 - 1.59477i) q^{4} +(-1.13197 - 0.558226i) q^{5} +(-2.28761 + 0.875702i) q^{6} +(3.03857 + 3.95994i) q^{7} +(-0.524089 + 2.77945i) q^{8} +(2.94501 + 0.571771i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26628	+	0.629706i	−0.895397	+	0.445270i
							\(3\)	1.72410	+	0.165818i	0.995407	+	0.0957349i
\(5\)	−1.13197	−	0.558226i	−0.506232	−	0.249646i								0.171193	−	0.985237i	\(-0.445238\pi\)
							−0.677426	+	0.735591i	\(0.736904\pi\)
\(7\)	3.03857	+	3.95994i	1.14847	+	1.49672i	0.837615	+	0.546261i	\(0.183949\pi\)
							0.310857	+	0.950457i	\(0.399384\pi\)
\(11\)	1.97099	+	1.72851i	0.594275	+	0.521165i	0.902805	−	0.430049i	\(-0.141504\pi\)
							−0.308530	+	0.951214i	\(0.599837\pi\)
\(13\)	−0.913144	−	0.0598506i	−0.253261	−	0.0165996i	−0.0617560	−	0.998091i	\(-0.519670\pi\)
							−0.191505	+	0.981492i	\(0.561337\pi\)
\(17\)	−3.51679	−	3.51679i	−0.852946	−	0.852946i	0.137549	−	0.990495i	\(-0.456078\pi\)
							−0.990495	+	0.137549i	\(0.956078\pi\)
\(19\)	0.121778	−	0.182254i	0.0279379	−	0.0418120i	−0.817238	−	0.576301i	\(-0.804496\pi\)
							0.845176	+	0.534489i	\(0.179496\pi\)
\(23\)	−2.61217	+	3.40424i	−0.544674	+	0.709833i	−0.981875	−	0.189531i	\(-0.939303\pi\)
							0.437200	+	0.899364i	\(0.355970\pi\)
\(29\)	−0.276780	+	0.815366i	−0.0513967	+	0.151410i	−0.969597	−	0.244706i	\(-0.921309\pi\)
							0.918201	+	0.396115i	\(0.129642\pi\)
\(31\)	5.21295	+	9.02909i	0.936273	+	1.62167i	0.772348	+	0.635200i	\(0.219082\pi\)
							0.163926	+	0.986473i	\(0.447584\pi\)
\(37\)	4.70139	−	3.14137i	0.772903	−	0.516438i	−0.105464	−	0.994423i	\(-0.533633\pi\)
							0.878368	+	0.477986i	\(0.158633\pi\)
\(41\)	−4.50511	−	3.45689i	−0.703580	−	0.539876i	0.193851	−	0.981031i	\(-0.437902\pi\)
							−0.897431	+	0.441155i	\(0.854569\pi\)
\(43\)	1.18235	+	1.03690i	0.180307	+	0.158125i	0.744742	−	0.667352i	\(-0.232573\pi\)
							−0.564435	+	0.825478i	\(0.690906\pi\)
\(47\)	0.499351	+	1.86360i	0.0728379	+	0.271835i	0.992734	−	0.120327i	\(-0.0383942\pi\)
							−0.919896	+	0.392161i	\(0.871728\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.11.14	✓	1504
9.5	odd	6	inner	576.2.bl.a.203.47	yes	1504
64.35	odd	16	inner	576.2.bl.a.227.47	yes	1504
576.419	even	48	inner	576.2.bl.a.419.14	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.14	✓	1504	1.1	even	1	trivial
576.2.bl.a.203.47	yes	1504	9.5	odd	6	inner
576.2.bl.a.227.47	yes	1504	64.35	odd	16	inner
576.2.bl.a.419.14	yes	1504	576.419	even	48	inner