Embedded newform 336.3.bv.a.325.25

• Label: 336.3.bv.a.325.25
• 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			325.25
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.620287 - 1.90138i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-3.23049 + 2.35880i) q^{4} +(-3.50467 - 0.939075i) q^{5} +(-1.89013 - 2.90300i) q^{6} +(-1.65530 + 6.80147i) q^{7} +(6.48881 + 4.67925i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.620287	−	1.90138i	−0.310144	−	0.950690i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−3.50467	−	0.939075i	−0.700935	−	0.187815i								−0.109286	−	0.994010i	\(-0.534856\pi\)
							−0.591649	+	0.806195i	\(0.701523\pi\)
\(7\)	−1.65530	+	6.80147i	−0.236471	+	0.971638i
							\(11\)	14.5964	−	3.91109i	1.32695	−	0.355554i	0.475370	−	0.879786i	\(-0.342314\pi\)
0.851576	+	0.524232i	\(0.175647\pi\)
\(13\)	0.646638	+	0.646638i	0.0497414	+	0.0497414i	0.731540	−	0.681799i	\(-0.238802\pi\)
							−0.681799	+	0.731540i	\(0.738802\pi\)
\(17\)	17.0461	+	9.84156i	1.00271	+	0.578915i	0.909049	−	0.416689i	\(-0.136810\pi\)
							0.0936612	+	0.995604i	\(0.470143\pi\)
\(19\)	8.72952	−	32.5790i	0.459448	−	1.71468i	−0.215223	−	0.976565i	\(-0.569048\pi\)
							0.674671	−	0.738119i	\(-0.264286\pi\)
\(23\)	3.45699	−	1.99589i	0.150304	−	0.0867779i	−0.422962	−	0.906147i	\(-0.639010\pi\)
							0.573266	+	0.819370i	\(0.305676\pi\)
\(29\)	7.69205	−	7.69205i	0.265243	−	0.265243i	−0.561937	−	0.827180i	\(-0.689944\pi\)
							0.827180	+	0.561937i	\(0.189944\pi\)
\(31\)	44.7719	+	25.8491i	1.44426	+	0.833842i	0.998130	−	0.0611281i	\(-0.0194698\pi\)
							0.446126	+	0.894970i	\(0.352803\pi\)
\(37\)	−8.90502	+	33.2340i	−0.240676	+	0.898216i	0.734831	+	0.678250i	\(0.237261\pi\)
							−0.975508	+	0.219966i	\(0.929405\pi\)
\(41\)	57.9667			1.41382			0.706911	−	0.707302i	\(-0.250088\pi\)
							0.706911	+	0.707302i	\(0.250088\pi\)
\(43\)	4.23471	+	4.23471i	0.0984817	+	0.0984817i	0.754631	−	0.656149i	\(-0.227816\pi\)
							−0.656149	+	0.754631i	\(0.727816\pi\)
\(47\)	−50.4307	+	29.1162i	−1.07299	+	0.619494i	−0.928998	−	0.370084i	\(-0.879329\pi\)
							−0.143997	+	0.989578i	\(0.545995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.25	yes	256
7.5	odd	6	inner	336.3.bv.a.229.18	yes	256
16.13	even	4	inner	336.3.bv.a.157.18	yes	256
112.61	odd	12	inner	336.3.bv.a.61.25	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.25	✓	256	112.61	odd	12	inner
336.3.bv.a.157.18	yes	256	16.13	even	4	inner
336.3.bv.a.229.18	yes	256	7.5	odd	6	inner
336.3.bv.a.325.25	yes	256	1.1	even	1	trivial