Embedded newform 336.2.bs.a.19.13

• Label: 336.2.bs.a.19.13
• Level: $336$
• Weight: $2$
• Character: 336.19
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.13
Character	\(\chi\)	\(=\)	336.19
Dual form			336.2.bs.a.283.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.668833 - 1.24606i) q^{2} +(0.258819 - 0.965926i) q^{3} +(-1.10533 + 1.66681i) q^{4} +(0.513920 - 0.137704i) q^{5} +(-1.37671 + 0.323539i) q^{6} +(1.08086 + 2.41490i) q^{7} +(2.81622 + 0.262485i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 4 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.668833	−	1.24606i	−0.472936	−	0.881097i
							\(3\)	0.258819	−	0.965926i	0.149429	−	0.557678i
\(5\)	0.513920	−	0.137704i	0.229832	−	0.0615833i								−0.142065	−	0.989857i	\(-0.545374\pi\)
							0.371897	+	0.928274i	\(0.378708\pi\)
\(7\)	1.08086	+	2.41490i	0.408525	+	0.912747i
							\(11\)	1.67559	−	6.25340i	0.505210	−	1.88547i	0.0422154	−	0.999109i	\(-0.486558\pi\)
0.462995	−	0.886361i	\(-0.346775\pi\)
\(13\)	3.08799	−	3.08799i	0.856455	−	0.856455i	−0.134463	−	0.990919i	\(-0.542931\pi\)
							0.990919	+	0.134463i	\(0.0429310\pi\)
\(17\)	−1.95705	+	1.12990i	−0.474653	+	0.274041i	−0.718186	−	0.695852i	\(-0.755027\pi\)
							0.243532	+	0.969893i	\(0.421694\pi\)
\(19\)	0.117956	−	0.0316062i	0.0270610	−	0.00725097i	−0.245263	−	0.969457i	\(-0.578874\pi\)
							0.272324	+	0.962206i	\(0.412208\pi\)
\(23\)	2.16025	−	3.74167i	0.450444	−	0.780192i	−0.547970	−	0.836498i	\(-0.684599\pi\)
							0.998414	+	0.0563065i	\(0.0179324\pi\)
\(29\)	4.37123	+	4.37123i	0.811717	+	0.811717i	0.984891	−	0.173174i	\(-0.0554023\pi\)
							−0.173174	+	0.984891i	\(0.555402\pi\)
\(31\)	−1.25747	−	2.17799i	−0.225848	−	0.391179i	0.730726	−	0.682671i	\(-0.239182\pi\)
							−0.956573	+	0.291492i	\(0.905848\pi\)
\(37\)	0.554721	+	2.07025i	0.0911956	+	0.340346i	0.996415	−	0.0845975i	\(-0.0269604\pi\)
							−0.905220	+	0.424944i	\(0.860294\pi\)
\(41\)	8.63061			1.34787			0.673937	−	0.738789i	\(-0.264602\pi\)
							0.673937	+	0.738789i	\(0.264602\pi\)
\(43\)	−2.21407	−	2.21407i	−0.337643	−	0.337643i	0.517836	−	0.855480i	\(-0.326738\pi\)
							−0.855480	+	0.517836i	\(0.826738\pi\)
\(47\)	4.58222	−	7.93663i	0.668385	−	1.15768i	−0.309970	−	0.950746i	\(-0.600319\pi\)
							0.978356	−	0.206931i	\(-0.0663475\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bs.a.19.13	✓	128
7.3	odd	6	inner	336.2.bs.a.115.32	yes	128
16.11	odd	4	inner	336.2.bs.a.187.32	yes	128
112.59	even	12	inner	336.2.bs.a.283.13	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bs.a.19.13	✓	128	1.1	even	1	trivial
336.2.bs.a.115.32	yes	128	7.3	odd	6	inner
336.2.bs.a.187.32	yes	128	16.11	odd	4	inner
336.2.bs.a.283.13	yes	128	112.59	even	12	inner