Embedded newform 576.2.bl.a.11.12

• Label: 576.2.bl.a.11.12
• Level: $576$
• Weight: $2$
• Character: 576.11
• 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			11.12
Character	\(\chi\)	\(=\)	576.11
Dual form			576.2.bl.a.419.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30367 - 0.548138i) q^{2} +(1.29483 - 1.15039i) q^{3} +(1.39909 + 1.42918i) q^{4} +(-1.03041 - 0.508143i) q^{5} +(-2.31860 + 0.789982i) q^{6} +(1.75332 + 2.28498i) q^{7} +(-1.04056 - 2.63006i) q^{8} +(0.353186 - 2.97914i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30367	−	0.548138i	−0.921831	−	0.387592i
							\(3\)	1.29483	−	1.15039i	0.747572	−	0.664180i
\(5\)	−1.03041	−	0.508143i	−0.460814	−	0.227248i								0.197037	−	0.980396i	\(-0.436868\pi\)
							−0.657852	+	0.753148i	\(0.728535\pi\)
\(7\)	1.75332	+	2.28498i	0.662694	+	0.863640i	0.996885	−	0.0788654i	\(-0.0251297\pi\)
							−0.334191	+	0.942505i	\(0.608463\pi\)
\(11\)	−3.86405	−	3.38868i	−1.16505	−	1.02172i	−0.999368	−	0.0355459i	\(-0.988683\pi\)
							−0.165686	−	0.986179i	\(-0.552984\pi\)
\(13\)	−5.56641	−	0.364842i	−1.54385	−	0.101189i	−0.730477	−	0.682938i	\(-0.760702\pi\)
							−0.813368	+	0.581749i	\(0.802369\pi\)
\(17\)	0.422740	+	0.422740i	0.102529	+	0.102529i	0.756511	−	0.653981i	\(-0.226902\pi\)
							−0.653981	+	0.756511i	\(0.726902\pi\)
\(19\)	4.18242	−	6.25943i	0.959513	−	1.43601i	0.0604607	−	0.998171i	\(-0.480743\pi\)
							0.899052	−	0.437842i	\(-0.144257\pi\)
\(23\)	2.49143	−	3.24690i	0.519500	−	0.677025i	−0.457759	−	0.889076i	\(-0.651348\pi\)
							0.977259	+	0.212051i	\(0.0680144\pi\)
\(29\)	1.73601	−	5.11412i	0.322369	−	0.949669i	−0.658089	−	0.752940i	\(-0.728635\pi\)
							0.980458	−	0.196728i	\(-0.0630317\pi\)
\(31\)	0.836001	+	1.44800i	0.150150	+	0.260068i	0.931283	−	0.364298i	\(-0.118691\pi\)
							−0.781132	+	0.624365i	\(0.785358\pi\)
\(37\)	6.04294	−	4.03776i	0.993452	−	0.663804i	0.0511935	−	0.998689i	\(-0.483697\pi\)
							0.942259	+	0.334885i	\(0.108697\pi\)
\(41\)	4.08599	+	3.13529i	0.638124	+	0.489650i	0.876423	−	0.481542i	\(-0.159923\pi\)
							−0.238299	+	0.971192i	\(0.576590\pi\)
\(43\)	−3.22453	−	2.82784i	−0.491737	−	0.431241i	0.377225	−	0.926122i	\(-0.376879\pi\)
							−0.868961	+	0.494880i	\(0.835212\pi\)
\(47\)	2.07557	+	7.74614i	0.302753	+	1.12989i	0.934862	+	0.355011i	\(0.115523\pi\)
							−0.632109	+	0.774880i	\(0.717810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.11.12	✓	1504
9.5	odd	6	inner	576.2.bl.a.203.21	yes	1504
64.35	odd	16	inner	576.2.bl.a.227.21	yes	1504
576.419	even	48	inner	576.2.bl.a.419.12	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.12	✓	1504	1.1	even	1	trivial
576.2.bl.a.203.21	yes	1504	9.5	odd	6	inner
576.2.bl.a.227.21	yes	1504	64.35	odd	16	inner
576.2.bl.a.419.12	yes	1504	576.419	even	48	inner