Embedded newform 336.3.bv.a.325.28

• Label: 336.3.bv.a.325.28
• Level: $336$
• Weight: $3$
• Character: 336.325
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			325.28
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.395331 - 1.96054i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(-3.68743 + 1.55013i) q^{4} +(-1.60095 - 0.428972i) q^{5} +(1.54029 + 3.10282i) q^{6} +(6.62188 - 2.26953i) q^{7} +(4.49684 + 6.61653i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.395331	−	1.96054i	−0.197666	−	0.980269i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−1.60095	−	0.428972i	−0.320189	−	0.0857945i								0.0951446	−	0.995463i	\(-0.469669\pi\)
							−0.415334	+	0.909669i	\(0.636335\pi\)
\(7\)	6.62188	−	2.26953i	0.945982	−	0.324218i
							\(11\)	3.06857	−	0.822220i	0.278960	−	0.0747472i	−0.116626	−	0.993176i	\(-0.537208\pi\)
0.395587	+	0.918429i	\(0.370541\pi\)
\(13\)	0.428517	+	0.428517i	0.0329629	+	0.0329629i	0.723396	−	0.690433i	\(-0.242580\pi\)
							−0.690433	+	0.723396i	\(0.742580\pi\)
\(17\)	1.13602	+	0.655879i	0.0668244	+	0.0385811i	0.533040	−	0.846090i	\(-0.321050\pi\)
							−0.466216	+	0.884671i	\(0.654383\pi\)
\(19\)	7.33463	−	27.3732i	0.386033	−	1.44070i	−0.450498	−	0.892777i	\(-0.648754\pi\)
							0.836531	−	0.547919i	\(-0.184580\pi\)
\(23\)	−35.7841	+	20.6599i	−1.55583	+	0.898258i	−0.558180	+	0.829720i	\(0.688500\pi\)
							−0.997649	+	0.0685381i	\(0.978167\pi\)
\(29\)	33.9704	−	33.9704i	1.17139	−	1.17139i	0.189517	−	0.981878i	\(-0.439308\pi\)
							0.981878	−	0.189517i	\(-0.0606921\pi\)
\(31\)	−29.9130	−	17.2703i	−0.964935	−	0.557105i	−0.0672466	−	0.997736i	\(-0.521421\pi\)
							−0.897688	+	0.440631i	\(0.854755\pi\)
\(37\)	18.3138	−	68.3479i	0.494967	−	1.84724i	−0.0352490	−	0.999379i	\(-0.511222\pi\)
							0.530216	−	0.847863i	\(-0.322111\pi\)
\(41\)	−63.5691			−1.55047			−0.775233	−	0.631675i	\(-0.782368\pi\)
							−0.775233	+	0.631675i	\(0.782368\pi\)
\(43\)	31.4733	+	31.4733i	0.731937	+	0.731937i	0.971003	−	0.239067i	\(-0.0768414\pi\)
							−0.239067	+	0.971003i	\(0.576841\pi\)
\(47\)	15.2172	−	8.78565i	0.323770	−	0.186929i	−0.329302	−	0.944225i	\(-0.606813\pi\)
							0.653072	+	0.757296i	\(0.273480\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.28	yes	256
7.5	odd	6	inner	336.3.bv.a.229.14	yes	256
16.13	even	4	inner	336.3.bv.a.157.14	yes	256
112.61	odd	12	inner	336.3.bv.a.61.28	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.28	✓	256	112.61	odd	12	inner
336.3.bv.a.157.14	yes	256	16.13	even	4	inner
336.3.bv.a.229.14	yes	256	7.5	odd	6	inner
336.3.bv.a.325.28	yes	256	1.1	even	1	trivial