Embedded newform 576.2.bm.a.517.50

• Label: 576.2.bm.a.517.50
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.50
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.50

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.109418 - 1.40997i) q^{2} +(-0.0225197 - 1.73190i) q^{3} +(-1.97606 - 0.308553i) q^{4} +(-2.28142 + 2.60146i) q^{5} +(-2.44440 - 0.157749i) q^{6} +(4.18621 - 0.551125i) q^{7} +(-0.651268 + 2.75243i) q^{8} +(-2.99899 + 0.0780039i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.109418	−	1.40997i	0.0773702	−	0.997002i
							\(3\)	−0.0225197	−	1.73190i	−0.0130017	−	0.999915i
\(5\)	−2.28142	+	2.60146i	−1.02028	+	1.16341i								−0.0338160	+	0.999428i	\(0.510766\pi\)
							−0.986464	+	0.163978i	\(0.947567\pi\)
\(7\)	4.18621	−	0.551125i	1.58224	−	0.208306i	0.712447	−	0.701725i	\(-0.247587\pi\)
							0.869791	+	0.493420i	\(0.164253\pi\)
\(11\)	−1.27415	+	2.58372i	−0.384170	+	0.779020i	−0.999960	−	0.00888988i	\(-0.997170\pi\)
							0.615790	+	0.787910i	\(0.288837\pi\)
\(13\)	−2.06007	+	6.06878i	−0.571362	+	1.68318i	0.148375	+	0.988931i	\(0.452596\pi\)
							−0.719737	+	0.694247i	\(0.755738\pi\)
\(17\)	−3.99244	+	3.99244i	−0.968310	+	0.968310i	−0.999513	−	0.0312035i	\(-0.990066\pi\)
							0.0312035	+	0.999513i	\(0.490066\pi\)
\(19\)	0.252318	+	1.26849i	0.0578856	+	0.291011i	0.998875	−	0.0474261i	\(-0.0151019\pi\)
							−0.940989	+	0.338437i	\(0.890102\pi\)
\(23\)	0.851387	−	6.46692i	0.177526	−	1.34845i	−0.638138	−	0.769922i	\(-0.720295\pi\)
							0.815665	−	0.578525i	\(-0.196372\pi\)
\(29\)	1.06532	+	0.0698247i	0.197825	+	0.0129661i	0.163993	−	0.986461i	\(-0.447562\pi\)
							0.0338314	+	0.999428i	\(0.489229\pi\)
\(31\)	−1.47312	−	0.850507i	−0.264580	−	0.152756i	0.361842	−	0.932240i	\(-0.382148\pi\)
							−0.626422	+	0.779484i	\(0.715481\pi\)
\(37\)	0.401559	−	2.01877i	0.0660159	−	0.331884i	−0.933637	−	0.358220i	\(-0.883384\pi\)
							0.999653	+	0.0263355i	\(0.00838382\pi\)
\(41\)	−0.989500	+	7.51600i	−0.154534	+	1.17380i	0.721801	+	0.692100i	\(0.243314\pi\)
							−0.876335	+	0.481701i	\(0.840019\pi\)
\(43\)	0.927365	+	0.457326i	0.141422	+	0.0697415i	0.511623	−	0.859210i	\(-0.329045\pi\)
							−0.370201	+	0.928952i	\(0.620711\pi\)
\(47\)	−3.04900	−	0.816978i	−0.444743	−	0.119169i	0.0294956	−	0.999565i	\(-0.490610\pi\)
							−0.474239	+	0.880396i	\(0.657277\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.50	yes	1504
9.7	even	3	inner	576.2.bm.a.133.14	yes	1504
64.13	even	16	inner	576.2.bm.a.13.14	✓	1504
576.205	even	48	inner	576.2.bm.a.205.50	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.14	✓	1504	64.13	even	16	inner
576.2.bm.a.133.14	yes	1504	9.7	even	3	inner
576.2.bm.a.205.50	yes	1504	576.205	even	48	inner
576.2.bm.a.517.50	yes	1504	1.1	even	1	trivial