Embedded newform 576.2.bl.a.419.12

• Label: 576.2.bl.a.419.12
• Level: $576$
• Weight: $2$
• Character: 576.419
• 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			419.12
Character	\(\chi\)	\(=\)	576.419
Dual form			576.2.bl.a.11.12

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{11}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.12	✓	1504	576.11	even	48	inner
576.2.bl.a.203.21	yes	1504	64.11	odd	16	inner
576.2.bl.a.227.21	yes	1504	9.2	odd	6	inner
576.2.bl.a.419.12	yes	1504	1.1	even	1	trivial