Embedded newform 336.3.bv.a.325.55

• Label: 336.3.bv.a.325.55
• 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			325.55
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78653 + 0.899069i) q^{2} +(1.67303 - 0.448288i) q^{3} +(2.38335 + 3.21242i) q^{4} +(3.93930 + 1.05553i) q^{5} +(3.39196 + 0.703295i) q^{6} +(-6.08772 + 3.45537i) q^{7} +(1.36972 + 7.88187i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.78653	+	0.899069i	0.893263	+	0.449535i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	3.93930	+	1.05553i	0.787860	+	0.211107i								0.630247	−	0.776394i	\(-0.282953\pi\)
							0.157613	+	0.987501i	\(0.449620\pi\)
\(7\)	−6.08772	+	3.45537i	−0.869675	+	0.493625i
							\(11\)	−14.3060	+	3.83328i	−1.30055	+	0.348480i	−0.841657	−	0.540013i	\(-0.818419\pi\)
−0.458889	+	0.888493i	\(0.651753\pi\)
\(13\)	15.4751	+	15.4751i	1.19039	+	1.19039i	0.976958	+	0.213433i	\(0.0684644\pi\)
							0.213433	+	0.976958i	\(0.431536\pi\)
\(17\)	8.85147	+	5.11040i	0.520675	+	0.300612i	0.737211	−	0.675663i	\(-0.236143\pi\)
							−0.216536	+	0.976275i	\(0.569476\pi\)
\(19\)	8.79016	−	32.8053i	0.462640	−	1.72660i	−0.201957	−	0.979394i	\(-0.564730\pi\)
							0.664597	−	0.747202i	\(-0.268603\pi\)
\(23\)	19.6636	−	11.3528i	0.854938	−	0.493599i	−0.00737565	−	0.999973i	\(-0.502348\pi\)
							0.862314	+	0.506374i	\(0.169014\pi\)
\(29\)	28.0441	−	28.0441i	0.967037	−	0.967037i	−0.0324369	−	0.999474i	\(-0.510327\pi\)
							0.999474	+	0.0324369i	\(0.0103268\pi\)
\(31\)	−30.2487	−	17.4641i	−0.975764	−	0.563357i	−0.0747752	−	0.997200i	\(-0.523824\pi\)
							−0.900988	+	0.433843i	\(0.857157\pi\)
\(37\)	6.41714	−	23.9491i	0.173436	−	0.647273i	−0.823376	−	0.567496i	\(-0.807912\pi\)
							0.996813	−	0.0797773i	\(-0.0254209\pi\)
\(41\)	−28.3560			−0.691609			−0.345805	−	0.938307i	\(-0.612394\pi\)
							−0.345805	+	0.938307i	\(0.612394\pi\)
\(43\)	44.6040	+	44.6040i	1.03730	+	1.03730i	0.999277	+	0.0380257i	\(0.0121069\pi\)
							0.0380257	+	0.999277i	\(0.487893\pi\)
\(47\)	−9.83669	+	5.67921i	−0.209291	+	0.120834i	−0.600982	−	0.799263i	\(-0.705224\pi\)
							0.391691	+	0.920097i	\(0.371890\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.55	yes	256
7.5	odd	6	inner	336.3.bv.a.229.31	yes	256
16.13	even	4	inner	336.3.bv.a.157.31	yes	256
112.61	odd	12	inner	336.3.bv.a.61.55	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.55	✓	256	112.61	odd	12	inner
336.3.bv.a.157.31	yes	256	16.13	even	4	inner
336.3.bv.a.229.31	yes	256	7.5	odd	6	inner
336.3.bv.a.325.55	yes	256	1.1	even	1	trivial