Embedded newform 384.2.w.a.131.59

• Label: 384.2.w.a.131.59
• Level: $384$
• Weight: $2$
• Character: 384.131
• Analytic conductor: $3.066$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.w (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.06625543762\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			131.59
Character	\(\chi\)	\(=\)	384.131
Dual form			384.2.w.a.299.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38248 - 0.297929i) q^{2} +(1.72957 - 0.0927440i) q^{3} +(1.82248 - 0.823758i) q^{4} +(-0.606287 + 1.99866i) q^{5} +(2.36345 - 0.643503i) q^{6} +(-0.441862 - 2.22139i) q^{7} +(2.27411 - 1.68179i) q^{8} +(2.98280 - 0.320814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 16 q^{3} - 32 q^{4} - 16 q^{6} - 32 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38248	−	0.297929i	0.977558	−	0.210667i
							\(3\)	1.72957	−	0.0927440i	0.998565	−	0.0535458i
\(5\)	−0.606287	+	1.99866i	−0.271140	+	0.893828i								0.710599	+	0.703597i	\(0.248424\pi\)
							−0.981739	+	0.190232i	\(0.939076\pi\)
\(7\)	−0.441862	−	2.22139i	−0.167008	−	0.839607i	−0.969904	−	0.243488i	\(-0.921709\pi\)
							0.802896	−	0.596119i	\(-0.203291\pi\)
\(11\)	−4.48085	+	0.441325i	−1.35103	+	0.133065i	−0.747491	−	0.664271i	\(-0.768742\pi\)
							−0.603536	+	0.797336i	\(0.706242\pi\)
\(13\)	−0.494698	−	1.63080i	−0.137205	−	0.452303i	0.861231	−	0.508213i	\(-0.169694\pi\)
							−0.998436	+	0.0559106i	\(0.982194\pi\)
\(17\)	2.01633	+	0.835190i	0.489031	+	0.202563i	0.613553	−	0.789653i	\(-0.289740\pi\)
							−0.124522	+	0.992217i	\(0.539740\pi\)
\(19\)	−4.00951	+	7.50127i	−0.919845	+	1.72091i	−0.266522	+	0.963829i	\(0.585875\pi\)
							−0.653323	+	0.757080i	\(0.726625\pi\)
\(23\)	0.408269	+	0.611018i	0.0851300	+	0.127406i	0.871591	−	0.490234i	\(-0.163089\pi\)
							−0.786461	+	0.617640i	\(0.788089\pi\)
\(29\)	−5.82767	−	0.573975i	−1.08217	−	0.106585i	−0.458822	−	0.888528i	\(-0.651728\pi\)
							−0.623349	+	0.781944i	\(0.714228\pi\)
\(31\)	0.361193	+	0.361193i	0.0648723	+	0.0648723i	0.738799	−	0.673926i	\(-0.235393\pi\)
							−0.673926	+	0.738799i	\(0.735393\pi\)
\(37\)	−3.66025	−	6.84785i	−0.601742	−	1.12578i	−0.979797	−	0.199996i	\(-0.935907\pi\)
							0.378055	−	0.925783i	\(-0.376593\pi\)
\(41\)	5.77154	+	8.63772i	0.901363	+	1.34898i	0.936893	+	0.349616i	\(0.113688\pi\)
							−0.0355305	+	0.999369i	\(0.511312\pi\)
\(43\)	−8.33382	−	6.83939i	−1.27090	−	1.04300i	−0.996688	−	0.0813264i	\(-0.974084\pi\)
							−0.274208	−	0.961670i	\(-0.588416\pi\)
\(47\)	−11.8375	−	4.90324i	−1.72667	−	0.715211i	−0.999589	−	0.0286610i	\(-0.990876\pi\)
							−0.727083	−	0.686550i	\(-0.759124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.2.w.a.131.59	yes	992
3.2	odd	2	inner	384.2.w.a.131.4	✓	992
128.43	odd	32	inner	384.2.w.a.299.4	yes	992
384.299	even	32	inner	384.2.w.a.299.59	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.w.a.131.4	✓	992	3.2	odd	2	inner
384.2.w.a.131.59	yes	992	1.1	even	1	trivial
384.2.w.a.299.4	yes	992	128.43	odd	32	inner
384.2.w.a.299.59	yes	992	384.299	even	32	inner