Embedded newform 576.2.bl.a.275.9

• Label: 576.2.bl.a.275.9
• Level: $576$
• Weight: $2$
• Character: 576.275
• 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			275.9
Character	\(\chi\)	\(=\)	576.275
Dual form			576.2.bl.a.155.9

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{7}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.41	✓	1504	9.2	odd	6	inner
576.2.bl.a.155.9	yes	1504	576.155	even	48	inner
576.2.bl.a.275.9	yes	1504	1.1	even	1	trivial
576.2.bl.a.347.41	yes	1504	64.27	odd	16	inner