Embedded newform 576.2.bl.a.155.74

• Label: 576.2.bl.a.155.74
• Level: $576$
• Weight: $2$
• Character: 576.155
• 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			155.74
Character	\(\chi\)	\(=\)	576.155
Dual form			576.2.bl.a.275.74

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.14228 + 0.833780i) q^{2} +(-1.72760 - 0.124138i) q^{3} +(0.609622 + 1.90483i) q^{4} +(-0.876607 - 2.58240i) q^{5} +(-1.86990 - 1.58224i) q^{6} +(-0.217749 + 0.167085i) q^{7} +(-0.891845 + 2.68414i) q^{8} +(2.96918 + 0.428920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.14228	+	0.833780i	0.807716	+	0.589571i
							\(3\)	−1.72760	−	0.124138i	−0.997428	−	0.0716709i
\(5\)	−0.876607	−	2.58240i	−0.392031	−	1.15489i								−0.946506	−	0.322685i	\(-0.895415\pi\)
							0.554476	−	0.832200i	\(-0.312919\pi\)
\(7\)	−0.217749	+	0.167085i	−0.0823015	+	0.0631522i	−0.649093	−	0.760709i	\(-0.724851\pi\)
							0.566791	+	0.823861i	\(0.308185\pi\)
\(11\)	0.277841	+	4.23904i	0.0837722	+	1.27812i	0.808258	+	0.588828i	\(0.200410\pi\)
							−0.724486	+	0.689289i	\(0.757923\pi\)
\(13\)	−1.17967	+	1.03454i	−0.327180	+	0.286929i	−0.807139	−	0.590362i	\(-0.798985\pi\)
							0.479958	+	0.877291i	\(0.340652\pi\)
\(17\)	4.84638	+	4.84638i	1.17542	+	1.17542i	0.980898	+	0.194522i	\(0.0623154\pi\)
							0.194522	+	0.980898i	\(0.437685\pi\)
\(19\)	0.997833	+	5.01645i	0.228919	+	1.15085i	0.908704	+	0.417441i	\(0.137073\pi\)
							−0.679785	+	0.733411i	\(0.737927\pi\)
\(23\)	4.20619	+	3.22752i	0.877051	+	0.672985i	0.945835	−	0.324646i	\(-0.105245\pi\)
							−0.0687842	+	0.997632i	\(0.521912\pi\)
\(29\)	−1.57743	−	3.19871i	−0.292922	−	0.593986i	0.699811	−	0.714328i	\(-0.253268\pi\)
							−0.992733	+	0.120342i	\(0.961601\pi\)
\(31\)	−3.23290	−	5.59954i	−0.580645	−	1.00571i	−0.995403	−	0.0957751i	\(-0.969467\pi\)
							0.414758	−	0.909932i	\(-0.363866\pi\)
\(37\)	−4.32593	−	0.860480i	−0.711178	−	0.141462i	−0.173772	−	0.984786i	\(-0.555596\pi\)
							−0.537406	+	0.843324i	\(0.680596\pi\)
\(41\)	3.91608	−	5.10353i	0.611589	−	0.797038i	−0.380203	−	0.924903i	\(-0.624146\pi\)
							0.991792	+	0.127865i	\(0.0408125\pi\)
\(43\)	0.623930	+	9.51932i	0.0951484	+	1.45168i	0.732320	+	0.680961i	\(0.238438\pi\)
							−0.637171	+	0.770722i	\(0.719896\pi\)
\(47\)	0.905576	+	3.37966i	0.132092	+	0.492974i	0.999993	−	0.00376356i	\(-0.00119798\pi\)
							−0.867901	+	0.496737i	\(0.834531\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.155.74	yes	1504
9.5	odd	6	inner	576.2.bl.a.347.83	yes	1504
64.19	odd	16	inner	576.2.bl.a.83.83	✓	1504
576.275	even	48	inner	576.2.bl.a.275.74	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.83	✓	1504	64.19	odd	16	inner
576.2.bl.a.155.74	yes	1504	1.1	even	1	trivial
576.2.bl.a.275.74	yes	1504	576.275	even	48	inner
576.2.bl.a.347.83	yes	1504	9.5	odd	6	inner