Embedded newform 576.2.bm.a.517.35

• Label: 576.2.bm.a.517.35
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.35
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.558490 - 1.29926i) q^{2} +(-0.103681 + 1.72894i) q^{3} +(-1.37618 + 1.45125i) q^{4} +(-0.832536 + 0.949325i) q^{5} +(2.30426 - 0.830889i) q^{6} +(-0.877285 + 0.115497i) q^{7} +(2.65414 + 0.977509i) q^{8} +(-2.97850 - 0.358517i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.558490	−	1.29926i	−0.394912	−	0.918719i
							\(3\)	−0.103681	+	1.72894i	−0.0598602	+	0.998207i
\(5\)	−0.832536	+	0.949325i	−0.372321	+	0.424551i								−0.907411	−	0.420244i	\(-0.861944\pi\)
							0.535090	+	0.844795i	\(0.320278\pi\)
\(7\)	−0.877285	+	0.115497i	−0.331583	+	0.0436537i	−0.294480	−	0.955658i	\(-0.595146\pi\)
							−0.0371030	+	0.999311i	\(0.511813\pi\)
\(11\)	−0.764844	+	1.55095i	−0.230609	+	0.467629i	−0.981163	−	0.193180i	\(-0.938120\pi\)
							0.750554	+	0.660809i	\(0.229787\pi\)
\(13\)	−0.0664904	+	0.195874i	−0.0184411	+	0.0543258i	−0.955761	−	0.294144i	\(-0.904965\pi\)
							0.937320	+	0.348470i	\(0.113299\pi\)
\(17\)	−1.23907	+	1.23907i	−0.300518	+	0.300518i	−0.841216	−	0.540699i	\(-0.818160\pi\)
							0.540699	+	0.841216i	\(0.318160\pi\)
\(19\)	−1.64561	−	8.27306i	−0.377530	−	1.89797i	−0.436490	−	0.899709i	\(-0.643779\pi\)
							0.0589603	−	0.998260i	\(-0.481221\pi\)
\(23\)	0.139552	−	1.06001i	0.0290987	−	0.221027i	−0.970717	−	0.240226i	\(-0.922778\pi\)
							0.999816	+	0.0191995i	\(0.00611176\pi\)
\(29\)	−7.86257	−	0.515340i	−1.46004	−	0.0956962i	−0.685433	−	0.728136i	\(-0.740387\pi\)
							−0.774610	+	0.632440i	\(0.782054\pi\)
\(31\)	−1.77296	−	1.02362i	−0.318432	−	0.183847i	0.332261	−	0.943187i	\(-0.392188\pi\)
							−0.650694	+	0.759340i	\(0.725522\pi\)
\(37\)	1.88482	−	9.47565i	0.309863	−	1.55779i	−0.441109	−	0.897454i	\(-0.645415\pi\)
							0.750972	−	0.660334i	\(-0.229585\pi\)
\(41\)	−0.845937	+	6.42553i	−0.132113	+	1.00350i	0.789052	+	0.614327i	\(0.210572\pi\)
							−0.921165	+	0.389173i	\(0.872761\pi\)
\(43\)	−2.65477	−	1.30919i	−0.404848	−	0.199649i	0.228452	−	0.973555i	\(-0.426634\pi\)
							−0.633300	+	0.773906i	\(0.718300\pi\)
\(47\)	−8.46311	−	2.26768i	−1.23447	−	0.330776i	−0.418153	−	0.908377i	\(-0.637322\pi\)
							−0.816319	+	0.577601i	\(0.803989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.35	yes	1504
9.7	even	3	inner	576.2.bm.a.133.28	yes	1504
64.13	even	16	inner	576.2.bm.a.13.28	✓	1504
576.205	even	48	inner	576.2.bm.a.205.35	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.28	✓	1504	64.13	even	16	inner
576.2.bm.a.133.28	yes	1504	9.7	even	3	inner
576.2.bm.a.205.35	yes	1504	576.205	even	48	inner
576.2.bm.a.517.35	yes	1504	1.1	even	1	trivial