Embedded newform 336.3.bv.a.157.14

• Label: 336.3.bv.a.157.14
• Level: $336$
• Weight: $3$
• Character: 336.157
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $256$
• 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			157.14
Character	\(\chi\)	\(=\)	336.157
Dual form			336.3.bv.a.229.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50021 - 1.32264i) q^{2} +(-0.448288 - 1.67303i) q^{3} +(0.501265 + 3.96847i) q^{4} +(-0.428972 + 1.60095i) q^{5} +(-1.54029 + 3.10282i) q^{6} +(-6.62188 + 2.26953i) q^{7} +(4.49684 - 6.61653i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50021	−	1.32264i	−0.750105	−	0.661318i
							\(3\)	−0.448288	−	1.67303i	−0.149429	−	0.557678i
\(5\)	−0.428972	+	1.60095i	−0.0857945	+	0.320189i								−0.995463	−	0.0951446i	\(-0.969669\pi\)
							0.909669	+	0.415334i	\(0.136335\pi\)
\(7\)	−6.62188	+	2.26953i	−0.945982	+	0.324218i
							\(11\)	−0.822220	−	3.06857i	−0.0747472	−	0.278960i	0.918429	−	0.395587i	\(-0.129459\pi\)
−0.993176	+	0.116626i	\(0.962792\pi\)
\(13\)	−0.428517	+	0.428517i	−0.0329629	+	0.0329629i	−0.723396	−	0.690433i	\(-0.757420\pi\)
							0.690433	+	0.723396i	\(0.257420\pi\)
\(17\)	1.13602	+	0.655879i	0.0668244	+	0.0385811i	0.533040	−	0.846090i	\(-0.321050\pi\)
							−0.466216	+	0.884671i	\(0.654383\pi\)
\(19\)	27.3732	+	7.33463i	1.44070	+	0.386033i	0.892777	−	0.450498i	\(-0.148754\pi\)
							0.547919	+	0.836531i	\(0.315420\pi\)
\(23\)	35.7841	−	20.6599i	1.55583	−	0.898258i	0.558180	−	0.829720i	\(-0.311500\pi\)
							0.997649	−	0.0685381i	\(-0.0218335\pi\)
\(29\)	33.9704	+	33.9704i	1.17139	+	1.17139i	0.981878	+	0.189517i	\(0.0606921\pi\)
							0.189517	+	0.981878i	\(0.439308\pi\)
\(31\)	−29.9130	−	17.2703i	−0.964935	−	0.557105i	−0.0672466	−	0.997736i	\(-0.521421\pi\)
							−0.897688	+	0.440631i	\(0.854755\pi\)
\(37\)	−68.3479	−	18.3138i	−1.84724	−	0.494967i	−0.847863	−	0.530216i	\(-0.822111\pi\)
							−0.999379	+	0.0352490i	\(0.988778\pi\)
\(41\)	63.5691			1.55047			0.775233	−	0.631675i	\(-0.217632\pi\)
							0.775233	+	0.631675i	\(0.217632\pi\)
\(43\)	31.4733	−	31.4733i	0.731937	−	0.731937i	−0.239067	−	0.971003i	\(-0.576841\pi\)
							0.971003	+	0.239067i	\(0.0768414\pi\)
\(47\)	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.157.14	yes	256
7.5	odd	6	inner	336.3.bv.a.61.28	✓	256
16.5	even	4	inner	336.3.bv.a.325.28	yes	256
112.5	odd	12	inner	336.3.bv.a.229.14	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.28	✓	256	7.5	odd	6	inner
336.3.bv.a.157.14	yes	256	1.1	even	1	trivial
336.3.bv.a.229.14	yes	256	112.5	odd	12	inner
336.3.bv.a.325.28	yes	256	16.5	even	4	inner