Embedded newform 576.2.bm.a.205.92

• Label: 576.2.bm.a.205.92
• 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			205.92
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.92

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40670 + 0.145621i) q^{2} +(1.66749 - 0.468500i) q^{3} +(1.95759 + 0.409689i) q^{4} +(-2.43544 - 2.77709i) q^{5} +(2.41387 - 0.416217i) q^{6} +(-1.08316 - 0.142600i) q^{7} +(2.69407 + 0.861374i) q^{8} +(2.56102 - 1.56243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40670	+	0.145621i	0.994685	+	0.102970i
							\(3\)	1.66749	−	0.468500i	0.962723	−	0.270489i
\(5\)	−2.43544	−	2.77709i	−1.08916	−	1.24195i								−0.968435	−	0.249266i	\(-0.919811\pi\)
							−0.120727	−	0.992686i	\(-0.538523\pi\)
\(7\)	−1.08316	−	0.142600i	−0.409395	−	0.0538979i	−0.0769836	−	0.997032i	\(-0.524529\pi\)
							−0.332411	+	0.943134i	\(0.607862\pi\)
\(11\)	−0.293719	−	0.595602i	−0.0885595	−	0.179581i	0.848087	−	0.529857i	\(-0.177754\pi\)
							−0.936646	+	0.350276i	\(0.886088\pi\)
\(13\)	−1.92052	−	5.65767i	−0.532656	−	1.56915i	−0.796531	−	0.604597i	\(-0.793334\pi\)
							0.263875	−	0.964557i	\(-0.414999\pi\)
\(17\)	5.07100	+	5.07100i	1.22990	+	1.22990i	0.964001	+	0.265897i	\(0.0856681\pi\)
							0.265897	+	0.964001i	\(0.414332\pi\)
\(19\)	−0.437622	+	2.20007i	−0.100397	+	0.504731i	0.897563	+	0.440887i	\(0.145336\pi\)
							−0.997960	+	0.0638442i	\(0.979664\pi\)
\(23\)	0.588002	+	4.46632i	0.122607	+	0.931292i	0.936596	+	0.350411i	\(0.113958\pi\)
							−0.813989	+	0.580880i	\(0.802708\pi\)
\(29\)	6.42431	−	0.421072i	1.19296	−	0.0781910i	0.544050	−	0.839053i	\(-0.316890\pi\)
							0.648914	+	0.760862i	\(0.275223\pi\)
\(31\)	−7.15139	+	4.12886i	−1.28443	+	0.741565i	−0.977655	−	0.210217i	\(-0.932583\pi\)
							−0.306774	+	0.951782i	\(0.599250\pi\)
\(37\)	0.377120	+	1.89591i	0.0619982	+	0.311686i	0.999324	−	0.0367687i	\(-0.0117065\pi\)
							−0.937326	+	0.348455i	\(0.886706\pi\)
\(41\)	0.0252889	+	0.192088i	0.00394946	+	0.0299991i	0.993306	−	0.115512i	\(-0.0368508\pi\)
							−0.989357	+	0.145511i	\(0.953517\pi\)
\(43\)	−8.29206	+	4.08919i	−1.26453	+	0.623596i	−0.945954	−	0.324302i	\(-0.894871\pi\)
							−0.318574	+	0.947898i	\(0.603204\pi\)
\(47\)	8.82308	−	2.36414i	1.28698	−	0.344845i	0.450466	−	0.892793i	\(-0.351258\pi\)
							0.836512	+	0.547949i	\(0.184591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.92	yes	1504
9.4	even	3	inner	576.2.bm.a.13.29	✓	1504
64.5	even	16	inner	576.2.bm.a.133.29	yes	1504
576.517	even	48	inner	576.2.bm.a.517.92	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.29	✓	1504	9.4	even	3	inner
576.2.bm.a.133.29	yes	1504	64.5	even	16	inner
576.2.bm.a.205.92	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.92	yes	1504	576.517	even	48	inner