Embedded newform 336.3.bv.a.325.31

• Label: 336.3.bv.a.325.31
• 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			325.31
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0448023 + 1.99950i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-3.99599 - 0.179164i) q^{4} +(-5.88967 - 1.57813i) q^{5} +(0.821395 + 3.36531i) q^{6} +(6.89224 + 1.22356i) q^{7} +(0.537268 - 7.98194i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0448023	+	1.99950i	−0.0224012	+	0.999749i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−5.88967	−	1.57813i	−1.17793	−	0.315627i								−0.383828	−	0.923405i	\(-0.625394\pi\)
							−0.794107	+	0.607778i	\(0.792061\pi\)
\(7\)	6.89224	+	1.22356i	0.984605	+	0.174794i
							\(11\)	−2.05038	+	0.549397i	−0.186398	+	0.0499452i	−0.350810	−	0.936446i	\(-0.614094\pi\)
0.164412	+	0.986392i	\(0.447427\pi\)
\(13\)	−11.9865	−	11.9865i	−0.922042	−	0.922042i	0.0751317	−	0.997174i	\(-0.476062\pi\)
							−0.997174	+	0.0751317i	\(0.976062\pi\)
\(17\)	−23.0008	−	13.2795i	−1.35299	−	0.781150i	−0.364323	−	0.931273i	\(-0.618700\pi\)
							−0.988667	+	0.150123i	\(0.952033\pi\)
\(19\)	5.00209	−	18.6680i	0.263268	−	0.982529i	−0.700034	−	0.714109i	\(-0.746832\pi\)
							0.963302	−	0.268420i	\(-0.0865013\pi\)
\(23\)	20.8786	−	12.0543i	0.907767	−	0.524099i	0.0280547	−	0.999606i	\(-0.491069\pi\)
							0.879712	+	0.475507i	\(0.157735\pi\)
\(29\)	30.2220	−	30.2220i	1.04214	−	1.04214i	0.0430643	−	0.999072i	\(-0.486288\pi\)
							0.999072	−	0.0430643i	\(-0.0137120\pi\)
\(31\)	50.5194	+	29.1674i	1.62966	+	0.940884i	0.984194	+	0.177094i	\(0.0566696\pi\)
							0.645465	+	0.763790i	\(0.276664\pi\)
\(37\)	9.65047	−	36.0160i	0.260823	−	0.973406i	−0.703934	−	0.710265i	\(-0.748575\pi\)
							0.964757	−	0.263141i	\(-0.0847584\pi\)
\(41\)	−69.3423			−1.69128			−0.845638	−	0.533756i	\(-0.820780\pi\)
							−0.845638	+	0.533756i	\(0.820780\pi\)
\(43\)	−13.4478	−	13.4478i	−0.312739	−	0.312739i	0.533231	−	0.845970i	\(-0.320978\pi\)
							−0.845970	+	0.533231i	\(0.820978\pi\)
\(47\)	−59.5104	+	34.3584i	−1.26618	+	0.731029i	−0.974263	−	0.225415i	\(-0.927626\pi\)
							−0.291916	+	0.956444i	\(0.594293\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.31	yes	256
7.5	odd	6	inner	336.3.bv.a.229.55	yes	256
16.13	even	4	inner	336.3.bv.a.157.55	yes	256
112.61	odd	12	inner	336.3.bv.a.61.31	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.31	✓	256	112.61	odd	12	inner
336.3.bv.a.157.55	yes	256	16.13	even	4	inner
336.3.bv.a.229.55	yes	256	7.5	odd	6	inner
336.3.bv.a.325.31	yes	256	1.1	even	1	trivial