Embedded newform 576.2.bm.a.205.35

• Label: 576.2.bm.a.205.35
• 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			205.35
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.35

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.28	✓	1504	9.4	even	3	inner
576.2.bm.a.133.28	yes	1504	64.5	even	16	inner
576.2.bm.a.205.35	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.35	yes	1504	576.517	even	48	inner