Embedded newform 336.3.bv.a.325.57

• Label: 336.3.bv.a.325.57
• 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			325.57
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.86764 + 0.715499i) q^{2} +(1.67303 - 0.448288i) q^{3} +(2.97612 + 2.67258i) q^{4} +(4.29165 + 1.14995i) q^{5} +(3.44536 + 0.359815i) q^{6} +(-5.40722 - 4.44545i) q^{7} +(3.64608 + 7.12082i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.86764	+	0.715499i	0.933818	+	0.357749i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	4.29165	+	1.14995i	0.858331	+	0.229989i								0.661035	−	0.750355i	\(-0.270117\pi\)
							0.197296	+	0.980344i	\(0.436784\pi\)
\(7\)	−5.40722	−	4.44545i	−0.772460	−	0.635064i
							\(11\)	19.1679	−	5.13603i	1.74254	−	0.466912i	0.759531	−	0.650472i	\(-0.225429\pi\)
0.983009	+	0.183560i	\(0.0587620\pi\)
\(13\)	−12.5693	−	12.5693i	−0.966873	−	0.966873i	0.0325960	−	0.999469i	\(-0.489623\pi\)
							−0.999469	+	0.0325960i	\(0.989623\pi\)
\(17\)	13.6204	+	7.86374i	0.801200	+	0.462573i	0.843891	−	0.536515i	\(-0.180260\pi\)
							−0.0426907	+	0.999088i	\(0.513593\pi\)
\(19\)	−7.64533	+	28.5328i	−0.402386	+	1.50172i	0.406440	+	0.913677i	\(0.366770\pi\)
							−0.808826	+	0.588048i	\(0.799897\pi\)
\(23\)	−32.0473	+	18.5025i	−1.39336	+	0.804456i	−0.993685	−	0.112201i	\(-0.964210\pi\)
							−0.399674	+	0.916658i	\(0.630877\pi\)
\(29\)	22.9651	−	22.9651i	0.791898	−	0.791898i	−0.189904	−	0.981803i	\(-0.560818\pi\)
							0.981803	+	0.189904i	\(0.0608178\pi\)
\(31\)	−6.11602	−	3.53109i	−0.197291	−	0.113906i	0.398100	−	0.917342i	\(-0.369670\pi\)
							−0.595391	+	0.803436i	\(0.703003\pi\)
\(37\)	6.62855	−	24.7381i	0.179150	−	0.668597i	−0.816657	−	0.577123i	\(-0.804175\pi\)
							0.995807	−	0.0914743i	\(-0.0291579\pi\)
\(41\)	−44.3999			−1.08293			−0.541463	−	0.840725i	\(-0.682129\pi\)
							−0.541463	+	0.840725i	\(0.682129\pi\)
\(43\)	−19.5921	−	19.5921i	−0.455630	−	0.455630i	0.441588	−	0.897218i	\(-0.354415\pi\)
							−0.897218	+	0.441588i	\(0.854415\pi\)
\(47\)	−37.1488	+	21.4479i	−0.790399	+	0.456337i	−0.840103	−	0.542427i	\(-0.817506\pi\)
							0.0497038	+	0.998764i	\(0.484172\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.57	yes	256
7.5	odd	6	inner	336.3.bv.a.229.29	yes	256
16.13	even	4	inner	336.3.bv.a.157.29	yes	256
112.61	odd	12	inner	336.3.bv.a.61.57	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.57	✓	256	112.61	odd	12	inner
336.3.bv.a.157.29	yes	256	16.13	even	4	inner
336.3.bv.a.229.29	yes	256	7.5	odd	6	inner
336.3.bv.a.325.57	yes	256	1.1	even	1	trivial