Embedded newform 336.3.bv.a.325.63

• Label: 336.3.bv.a.325.63
• 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			325.63
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.63

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99733 - 0.103331i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(3.97865 - 0.412773i) q^{4} +(-3.88908 - 1.04207i) q^{5} +(-3.29527 + 1.06825i) q^{6} +(-3.30454 + 6.17090i) q^{7} +(7.90401 - 1.23556i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99733	−	0.103331i	0.998664	−	0.0516656i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−3.88908	−	1.04207i	−0.777815	−	0.208415i								−0.151994	−	0.988381i	\(-0.548569\pi\)
							−0.625821	+	0.779966i	\(0.715236\pi\)
\(7\)	−3.30454	+	6.17090i	−0.472078	+	0.881557i
							\(11\)	11.0164	−	2.95183i	1.00149	−	0.268348i	0.279420	−	0.960169i	\(-0.409858\pi\)
0.722069	+	0.691821i	\(0.243191\pi\)
\(13\)	15.1005	+	15.1005i	1.16158	+	1.16158i	0.984130	+	0.177451i	\(0.0567850\pi\)
							0.177451	+	0.984130i	\(0.443215\pi\)
\(17\)	20.9787	+	12.1121i	1.23404	+	0.712475i	0.967870	−	0.251450i	\(-0.0809076\pi\)
							0.266173	+	0.963925i	\(0.414241\pi\)
\(19\)	−6.76426	+	25.2446i	−0.356014	+	1.32866i	0.523191	+	0.852216i	\(0.324742\pi\)
							−0.879204	+	0.476445i	\(0.841925\pi\)
\(23\)	7.31177	−	4.22145i	0.317903	−	0.183541i	−0.332555	−	0.943084i	\(-0.607911\pi\)
							0.650457	+	0.759543i	\(0.274577\pi\)
\(29\)	−18.5586	+	18.5586i	−0.639950	+	0.639950i	−0.950543	−	0.310593i	\(-0.899472\pi\)
							0.310593	+	0.950543i	\(0.399472\pi\)
\(31\)	−40.8554	−	23.5879i	−1.31792	−	0.760900i	−0.334524	−	0.942387i	\(-0.608575\pi\)
							−0.983393	+	0.181488i	\(0.941909\pi\)
\(37\)	14.6452	−	54.6565i	0.395815	−	1.47720i	−0.424572	−	0.905394i	\(-0.639575\pi\)
							0.820387	−	0.571809i	\(-0.193758\pi\)
\(41\)	51.7308			1.26173			0.630863	−	0.775894i	\(-0.282701\pi\)
							0.630863	+	0.775894i	\(0.282701\pi\)
\(43\)	10.1268	+	10.1268i	0.235508	+	0.235508i	0.814987	−	0.579479i	\(-0.196744\pi\)
							−0.579479	+	0.814987i	\(0.696744\pi\)
\(47\)	−34.3169	+	19.8129i	−0.730147	+	0.421550i	−0.818476	−	0.574541i	\(-0.805181\pi\)
							0.0883291	+	0.996091i	\(0.471847\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.63	yes	256
7.5	odd	6	inner	336.3.bv.a.229.22	yes	256
16.13	even	4	inner	336.3.bv.a.157.22	yes	256
112.61	odd	12	inner	336.3.bv.a.61.63	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.63	✓	256	112.61	odd	12	inner
336.3.bv.a.157.22	yes	256	16.13	even	4	inner
336.3.bv.a.229.22	yes	256	7.5	odd	6	inner
336.3.bv.a.325.63	yes	256	1.1	even	1	trivial