Embedded newform 576.2.bl.a.419.7

• Label: 576.2.bl.a.419.7
• 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			419.7
Character	\(\chi\)	\(=\)	576.419
Dual form			576.2.bl.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37786 - 0.318576i) q^{2} +(-0.139389 + 1.72643i) q^{3} +(1.79702 + 0.877909i) q^{4} +(-0.245576 + 0.121104i) q^{5} +(0.742059 - 2.33438i) q^{6} +(0.961138 - 1.25258i) q^{7} +(-2.19637 - 1.78213i) q^{8} +(-2.96114 - 0.481292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37786	−	0.318576i	−0.974297	−	0.225267i
							\(3\)	−0.139389	+	1.72643i	−0.0804763	+	0.996757i
\(5\)	−0.245576	+	0.121104i	−0.109825	+	0.0541596i								−0.496371	−	0.868110i	\(-0.665335\pi\)
							0.386547	+	0.922270i	\(0.373668\pi\)
\(7\)	0.961138	−	1.25258i	0.363276	−	0.473431i	−0.575578	−	0.817747i	\(-0.695223\pi\)
							0.938854	+	0.344317i	\(0.111889\pi\)
\(11\)	−2.50606	+	2.19776i	−0.755606	+	0.662648i	−0.947856	−	0.318697i	\(-0.896755\pi\)
							0.192251	+	0.981346i	\(0.438421\pi\)
\(13\)	0.468830	−	0.0307287i	0.130030	−	0.00852262i	−0.000249145	−	1.00000i	\(-0.500079\pi\)
							0.130279	+	0.991477i	\(0.458413\pi\)
\(17\)	−4.48441	+	4.48441i	−1.08763	+	1.08763i	−0.0918566	+	0.995772i	\(0.529280\pi\)
							−0.995772	+	0.0918566i	\(0.970720\pi\)
\(19\)	−1.86202	−	2.78672i	−0.427178	−	0.639317i	0.553979	−	0.832531i	\(-0.313109\pi\)
							−0.981157	+	0.193214i	\(0.938109\pi\)
\(23\)	3.20641	+	4.17867i	0.668582	+	0.871313i	0.997324	−	0.0731064i	\(-0.0232913\pi\)
							−0.328742	+	0.944420i	\(0.606625\pi\)
\(29\)	−1.39807	−	4.11859i	−0.259615	−	0.764803i	−0.996091	−	0.0883379i	\(-0.971844\pi\)
							0.736475	−	0.676465i	\(-0.236489\pi\)
\(31\)	−2.61957	+	4.53723i	−0.470489	+	0.814911i	−0.999430	−	0.0337475i	\(-0.989256\pi\)
							0.528941	+	0.848658i	\(0.322589\pi\)
\(37\)	−7.67080	−	5.12546i	−1.26107	−	0.842621i	−0.268381	−	0.963313i	\(-0.586489\pi\)
							−0.992690	+	0.120692i	\(0.961489\pi\)
\(41\)	−4.44905	+	3.41387i	−0.694825	+	0.533158i	−0.894702	−	0.446663i	\(-0.852612\pi\)
							0.199878	+	0.979821i	\(0.435946\pi\)
\(43\)	4.90699	−	4.30332i	0.748309	−	0.656250i	−0.197747	−	0.980253i	\(-0.563362\pi\)
							0.946056	+	0.324004i	\(0.105029\pi\)
\(47\)	0.164065	−	0.612298i	0.0239313	−	0.0893129i	−0.952927	−	0.303199i	\(-0.901945\pi\)
							0.976859	+	0.213886i	\(0.0686121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.419.7	yes	1504
9.2	odd	6	inner	576.2.bl.a.227.38	yes	1504
64.11	odd	16	inner	576.2.bl.a.203.38	yes	1504
576.11	even	48	inner	576.2.bl.a.11.7	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.7	✓	1504	576.11	even	48	inner
576.2.bl.a.203.38	yes	1504	64.11	odd	16	inner
576.2.bl.a.227.38	yes	1504	9.2	odd	6	inner
576.2.bl.a.419.7	yes	1504	1.1	even	1	trivial