Embedded newform 336.3.bv.a.325.19

• Label: 336.3.bv.a.325.19
• 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			325.19
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22821 - 1.57845i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-0.982996 + 3.87733i) q^{4} +(-0.954978 - 0.255885i) q^{5} +(-2.76244 - 2.09020i) q^{6} +(-2.53920 - 6.52323i) q^{7} +(7.32750 - 3.21058i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22821	−	1.57845i	−0.614105	−	0.789224i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−0.954978	−	0.255885i	−0.190996	−	0.0511771i								0.162053	−	0.986782i	\(-0.448188\pi\)
							−0.353049	+	0.935605i	\(0.614855\pi\)
\(7\)	−2.53920	−	6.52323i	−0.362742	−	0.931889i
							\(11\)	10.8548	−	2.90853i	0.986799	−	0.264412i	0.270893	−	0.962609i	\(-0.412681\pi\)
0.715905	+	0.698197i	\(0.246014\pi\)
\(13\)	10.5874	+	10.5874i	0.814413	+	0.814413i	0.985292	−	0.170879i	\(-0.0546608\pi\)
							−0.170879	+	0.985292i	\(0.554661\pi\)
\(17\)	−24.5317	−	14.1634i	−1.44304	−	0.833140i	−0.444990	−	0.895536i	\(-0.646793\pi\)
							−0.998052	+	0.0623953i	\(0.980126\pi\)
\(19\)	−0.443951	+	1.65685i	−0.0233659	+	0.0872026i	−0.976624	−	0.214954i	\(-0.931040\pi\)
							0.953258	+	0.302157i	\(0.0977065\pi\)
\(23\)	25.3153	−	14.6158i	1.10066	−	0.635468i	0.164268	−	0.986416i	\(-0.447474\pi\)
							0.936395	+	0.350948i	\(0.114140\pi\)
\(29\)	25.9205	−	25.9205i	0.893811	−	0.893811i	−0.101069	−	0.994879i	\(-0.532226\pi\)
							0.994879	+	0.101069i	\(0.0322263\pi\)
\(31\)	−36.0049	−	20.7874i	−1.16145	−	0.670562i	−0.209797	−	0.977745i	\(-0.567280\pi\)
							−0.951650	+	0.307183i	\(0.900614\pi\)
\(37\)	7.08641	−	26.4469i	0.191525	−	0.714780i	−0.801614	−	0.597841i	\(-0.796025\pi\)
							0.993139	−	0.116939i	\(-0.0373081\pi\)
\(41\)	7.47809			0.182393			0.0911963	−	0.995833i	\(-0.470931\pi\)
							0.0911963	+	0.995833i	\(0.470931\pi\)
\(43\)	−60.4398	−	60.4398i	−1.40558	−	1.40558i	−0.780817	−	0.624760i	\(-0.785197\pi\)
							−0.624760	−	0.780817i	\(-0.714803\pi\)
\(47\)	49.7660	−	28.7324i	1.05885	−	0.611328i	0.133737	−	0.991017i	\(-0.457302\pi\)
							0.925114	+	0.379689i	\(0.123969\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.19	yes	256
7.5	odd	6	inner	336.3.bv.a.229.25	yes	256
16.13	even	4	inner	336.3.bv.a.157.25	yes	256
112.61	odd	12	inner	336.3.bv.a.61.19	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.19	✓	256	112.61	odd	12	inner
336.3.bv.a.157.25	yes	256	16.13	even	4	inner
336.3.bv.a.229.25	yes	256	7.5	odd	6	inner
336.3.bv.a.325.19	yes	256	1.1	even	1	trivial