Embedded newform 336.3.bv.a.325.18

• Label: 336.3.bv.a.325.18
• 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			325.18
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34692 - 1.47845i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-0.371607 + 3.98270i) q^{4} +(9.43009 + 2.52678i) q^{5} +(-2.91621 - 1.86968i) q^{6} +(-5.22069 + 4.66309i) q^{7} +(6.38874 - 4.81498i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34692	−	1.47845i	−0.673461	−	0.739223i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	9.43009	+	2.52678i	1.88602	+	0.505357i								0.999053	+	0.0435213i	\(0.0138576\pi\)
							0.886965	+	0.461836i	\(0.152809\pi\)
\(7\)	−5.22069	+	4.66309i	−0.745812	+	0.666156i
							\(11\)	1.67623	−	0.449144i	0.152384	−	0.0408312i	−0.181820	−	0.983332i	\(-0.558199\pi\)
0.334205	+	0.942501i	\(0.391532\pi\)
\(13\)	2.65629	+	2.65629i	0.204330	+	0.204330i	0.801852	−	0.597522i	\(-0.203848\pi\)
							−0.597522	+	0.801852i	\(0.703848\pi\)
\(17\)	10.0997	+	5.83106i	0.594100	+	0.343004i	0.766717	−	0.641985i	\(-0.221889\pi\)
							−0.172617	+	0.984989i	\(0.555222\pi\)
\(19\)	−2.63979	+	9.85182i	−0.138936	+	0.518517i	0.861014	+	0.508581i	\(0.169830\pi\)
							−0.999951	+	0.00993645i	\(0.996837\pi\)
\(23\)	−23.3211	+	13.4644i	−1.01396	+	0.585410i	−0.912349	−	0.409414i	\(-0.865733\pi\)
							−0.101611	+	0.994824i	\(0.532400\pi\)
\(29\)	−12.8631	+	12.8631i	−0.443557	+	0.443557i	−0.893205	−	0.449649i	\(-0.851549\pi\)
							0.449649	+	0.893205i	\(0.351549\pi\)
\(31\)	−3.79978	−	2.19380i	−0.122574	−	0.0707679i	0.437460	−	0.899238i	\(-0.355878\pi\)
							−0.560033	+	0.828470i	\(0.689212\pi\)
\(37\)	13.6392	−	50.9024i	0.368628	−	1.37574i	−0.493807	−	0.869572i	\(-0.664395\pi\)
							0.862435	−	0.506168i	\(-0.168938\pi\)
\(41\)	39.4047			0.961089			0.480545	−	0.876970i	\(-0.340439\pi\)
							0.480545	+	0.876970i	\(0.340439\pi\)
\(43\)	25.0711	+	25.0711i	0.583048	+	0.583048i	0.935740	−	0.352692i	\(-0.114734\pi\)
							−0.352692	+	0.935740i	\(0.614734\pi\)
\(47\)	66.4602	−	38.3708i	1.41405	−	0.816401i	0.418281	−	0.908318i	\(-0.362633\pi\)
							0.995767	+	0.0919168i	\(0.0292994\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.18	yes	256
7.5	odd	6	inner	336.3.bv.a.229.27	yes	256
16.13	even	4	inner	336.3.bv.a.157.27	yes	256
112.61	odd	12	inner	336.3.bv.a.61.18	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.18	✓	256	112.61	odd	12	inner
336.3.bv.a.157.27	yes	256	16.13	even	4	inner
336.3.bv.a.229.27	yes	256	7.5	odd	6	inner
336.3.bv.a.325.18	yes	256	1.1	even	1	trivial