Embedded newform 576.2.bl.a.155.9

• Label: 576.2.bl.a.155.9
• Level: $576$
• Weight: $2$
• Character: 576.155
• 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			155.9
Character	\(\chi\)	\(=\)	576.155
Dual form			576.2.bl.a.275.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35486 + 0.405404i) q^{2} +(0.810003 - 1.53098i) q^{3} +(1.67130 - 1.09853i) q^{4} +(-0.190356 - 0.560771i) q^{5} +(-0.476777 + 2.40264i) q^{6} +(-1.38116 + 1.05980i) q^{7} +(-1.81903 + 2.16591i) q^{8} +(-1.68779 - 2.48019i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35486	+	0.405404i	−0.958031	+	0.286664i
							\(3\)	0.810003	−	1.53098i	0.467655	−	0.883911i
\(5\)	−0.190356	−	0.560771i	−0.0851298	−	0.250784i								0.896026	−	0.444001i	\(-0.146441\pi\)
							−0.981156	+	0.193217i	\(0.938108\pi\)
\(7\)	−1.38116	+	1.05980i	−0.522029	+	0.400567i	−0.835886	−	0.548904i	\(-0.815045\pi\)
							0.313857	+	0.949470i	\(0.398379\pi\)
\(11\)	−0.168568	−	2.57185i	−0.0508251	−	0.775442i	−0.944866	−	0.327457i	\(-0.893808\pi\)
							0.894041	−	0.447985i	\(-0.147858\pi\)
\(13\)	−4.98069	+	4.36795i	−1.38139	+	1.21145i	−0.430248	+	0.902711i	\(0.641574\pi\)
							−0.951147	+	0.308739i	\(0.900093\pi\)
\(17\)	−3.24939	−	3.24939i	−0.788093	−	0.788093i	0.193088	−	0.981181i	\(-0.438150\pi\)
							−0.981181	+	0.193088i	\(0.938150\pi\)
\(19\)	0.0330210	+	0.166008i	0.00757555	+	0.0380848i	0.984388	−	0.176011i	\(-0.0563195\pi\)
							−0.976813	+	0.214096i	\(0.931319\pi\)
\(23\)	−5.65405	−	4.33850i	−1.17895	−	0.904641i	−0.182048	−	0.983290i	\(-0.558273\pi\)
							−0.996902	+	0.0786490i	\(0.974939\pi\)
\(29\)	−1.27224	−	2.57985i	−0.236249	−	0.479066i	0.746170	−	0.665756i	\(-0.231891\pi\)
							−0.982419	+	0.186690i	\(0.940224\pi\)
\(31\)	3.97508	+	6.88504i	0.713946	+	1.23659i	0.963365	+	0.268194i	\(0.0864270\pi\)
							−0.249419	+	0.968396i	\(0.580240\pi\)
\(37\)	−1.20923	−	0.240530i	−0.198796	−	0.0395429i	0.0946880	−	0.995507i	\(-0.469815\pi\)
							−0.293484	+	0.955964i	\(0.594815\pi\)
\(41\)	−3.83076	+	4.99235i	−0.598265	+	0.779674i	−0.990100	−	0.140362i	\(-0.955173\pi\)
							0.391836	+	0.920035i	\(0.371840\pi\)
\(43\)	−0.331939	−	5.06442i	−0.0506203	−	0.772316i	−0.945421	−	0.325851i	\(-0.894349\pi\)
							0.894801	−	0.446465i	\(-0.147317\pi\)
\(47\)	1.62319	+	6.05784i	0.236767	+	0.883627i	0.977344	+	0.211656i	\(0.0678856\pi\)
							−0.740577	+	0.671971i	\(0.765448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.155.9	yes	1504
9.5	odd	6	inner	576.2.bl.a.347.41	yes	1504
64.19	odd	16	inner	576.2.bl.a.83.41	✓	1504
576.275	even	48	inner	576.2.bl.a.275.9	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.41	✓	1504	64.19	odd	16	inner
576.2.bl.a.155.9	yes	1504	1.1	even	1	trivial
576.2.bl.a.275.9	yes	1504	576.275	even	48	inner
576.2.bl.a.347.41	yes	1504	9.5	odd	6	inner