Embedded newform 336.3.bv.a.157.22

• Label: 336.3.bv.a.157.22
• Level: $336$
• Weight: $3$
• Character: 336.157
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $256$
• 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			157.22
Character	\(\chi\)	\(=\)	336.157
Dual form			336.3.bv.a.229.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08815 + 1.67807i) q^{2} +(-0.448288 - 1.67303i) q^{3} +(-1.63185 - 3.65199i) q^{4} +(-1.04207 + 3.88908i) q^{5} +(3.29527 + 1.06825i) q^{6} +(3.30454 - 6.17090i) q^{7} +(7.90401 + 1.23556i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08815	+	1.67807i	−0.544076	+	0.839036i
							\(3\)	−0.448288	−	1.67303i	−0.149429	−	0.557678i
\(5\)	−1.04207	+	3.88908i	−0.208415	+	0.777815i								0.779966	+	0.625821i	\(0.215236\pi\)
							−0.988381	+	0.151994i	\(0.951431\pi\)
\(7\)	3.30454	−	6.17090i	0.472078	−	0.881557i
							\(11\)	−2.95183	−	11.0164i	−0.268348	−	1.00149i	−0.960169	−	0.279420i	\(-0.909858\pi\)
0.691821	−	0.722069i	\(-0.256809\pi\)
\(13\)	−15.1005	+	15.1005i	−1.16158	+	1.16158i	−0.177451	+	0.984130i	\(0.556785\pi\)
							−0.984130	+	0.177451i	\(0.943215\pi\)
\(17\)	20.9787	+	12.1121i	1.23404	+	0.712475i	0.967870	−	0.251450i	\(-0.0809076\pi\)
							0.266173	+	0.963925i	\(0.414241\pi\)
\(19\)	−25.2446	−	6.76426i	−1.32866	−	0.356014i	−0.476445	−	0.879204i	\(-0.658075\pi\)
							−0.852216	+	0.523191i	\(0.824742\pi\)
\(23\)	−7.31177	+	4.22145i	−0.317903	+	0.183541i	−0.650457	−	0.759543i	\(-0.725423\pi\)
							0.332555	+	0.943084i	\(0.392089\pi\)
\(29\)	−18.5586	−	18.5586i	−0.639950	−	0.639950i	0.310593	−	0.950543i	\(-0.399472\pi\)
							−0.950543	+	0.310593i	\(0.899472\pi\)
\(31\)	−40.8554	−	23.5879i	−1.31792	−	0.760900i	−0.334524	−	0.942387i	\(-0.608575\pi\)
							−0.983393	+	0.181488i	\(0.941909\pi\)
\(37\)	−54.6565	−	14.6452i	−1.47720	−	0.395815i	−0.571809	−	0.820387i	\(-0.693758\pi\)
							−0.905394	+	0.424572i	\(0.860425\pi\)
\(41\)	−51.7308			−1.26173			−0.630863	−	0.775894i	\(-0.717299\pi\)
							−0.630863	+	0.775894i	\(0.717299\pi\)
\(43\)	10.1268	−	10.1268i	0.235508	−	0.235508i	−0.579479	−	0.814987i	\(-0.696744\pi\)
							0.814987	+	0.579479i	\(0.196744\pi\)
\(47\)	−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.157.22	yes	256
7.5	odd	6	inner	336.3.bv.a.61.63	✓	256
16.5	even	4	inner	336.3.bv.a.325.63	yes	256
112.5	odd	12	inner	336.3.bv.a.229.22	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.63	✓	256	7.5	odd	6	inner
336.3.bv.a.157.22	yes	256	1.1	even	1	trivial
336.3.bv.a.229.22	yes	256	112.5	odd	12	inner
336.3.bv.a.325.63	yes	256	16.5	even	4	inner