Embedded newform 336.3.bv.a.325.6

• Label: 336.3.bv.a.325.6
• 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			325.6
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92787 + 0.532285i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(3.43335 - 2.05235i) q^{4} +(6.38763 + 1.71156i) q^{5} +(2.98677 - 1.75477i) q^{6} +(-0.663491 + 6.96848i) q^{7} +(-5.52660 + 5.78418i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92787	+	0.532285i	−0.963934	+	0.266143i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	6.38763	+	1.71156i	1.27753	+	0.342312i								0.832910	−	0.553409i	\(-0.186673\pi\)
							0.444617	+	0.895721i	\(0.353340\pi\)
\(7\)	−0.663491	+	6.96848i	−0.0947845	+	0.995498i
							\(11\)	−10.6201	+	2.84564i	−0.965463	+	0.258695i	−0.706911	−	0.707302i	\(-0.749912\pi\)
−0.258552	+	0.965997i	\(0.583245\pi\)
\(13\)	6.64888	+	6.64888i	0.511452	+	0.511452i	0.914971	−	0.403519i	\(-0.132213\pi\)
							−0.403519	+	0.914971i	\(0.632213\pi\)
\(17\)	18.2409	+	10.5314i	1.07300	+	0.619494i	0.928998	−	0.370084i	\(-0.120671\pi\)
							0.143997	+	0.989578i	\(0.454004\pi\)
\(19\)	6.60670	−	24.6566i	0.347721	−	1.29771i	−0.541679	−	0.840585i	\(-0.682211\pi\)
							0.889401	−	0.457128i	\(-0.151122\pi\)
\(23\)	−9.56486	+	5.52228i	−0.415864	+	0.240099i	−0.693306	−	0.720643i	\(-0.743847\pi\)
							0.277442	+	0.960742i	\(0.410513\pi\)
\(29\)	−33.5804	+	33.5804i	−1.15795	+	1.15795i	−0.173029	+	0.984917i	\(0.555356\pi\)
							−0.984917	+	0.173029i	\(0.944644\pi\)
\(31\)	−6.16653	−	3.56025i	−0.198920	−	0.114847i	0.397231	−	0.917718i	\(-0.369971\pi\)
							−0.596152	+	0.802872i	\(0.703304\pi\)
\(37\)	−2.39336	+	8.93214i	−0.0646854	+	0.241409i	−0.990697	−	0.136086i	\(-0.956548\pi\)
							0.926012	+	0.377495i	\(0.123214\pi\)
\(41\)	7.25275			0.176896			0.0884482	−	0.996081i	\(-0.471809\pi\)
							0.0884482	+	0.996081i	\(0.471809\pi\)
\(43\)	40.5299	+	40.5299i	0.942555	+	0.942555i	0.998437	−	0.0558823i	\(-0.0177972\pi\)
							−0.0558823	+	0.998437i	\(0.517797\pi\)
\(47\)	−63.4280	+	36.6202i	−1.34953	+	0.779153i	−0.988183	−	0.153278i	\(-0.951017\pi\)
							−0.361349	+	0.932431i	\(0.617684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.6	yes	256
7.5	odd	6	inner	336.3.bv.a.229.48	yes	256
16.13	even	4	inner	336.3.bv.a.157.48	yes	256
112.61	odd	12	inner	336.3.bv.a.61.6	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.6	✓	256	112.61	odd	12	inner
336.3.bv.a.157.48	yes	256	16.13	even	4	inner
336.3.bv.a.229.48	yes	256	7.5	odd	6	inner
336.3.bv.a.325.6	yes	256	1.1	even	1	trivial