Embedded newform 336.3.bv.a.325.48

• Label: 336.3.bv.a.325.48
• 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			325.48
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39314 - 1.43498i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(-0.118318 - 3.99825i) q^{4} +(2.73910 + 0.733940i) q^{5} +(-1.68749 + 3.02529i) q^{6} +(5.81243 + 3.90072i) q^{7} +(-5.90223 - 5.40034i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39314	−	1.43498i	0.696570	−	0.717488i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	2.73910	+	0.733940i	0.547820	+	0.146788i								0.522105	−	0.852881i	\(-0.325147\pi\)
							0.0257154	+	0.999669i	\(0.491814\pi\)
\(7\)	5.81243	+	3.90072i	0.830348	+	0.557246i
							\(11\)	6.89549	−	1.84764i	0.626863	−	0.167967i	0.0686171	−	0.997643i	\(-0.478141\pi\)
0.558246	+	0.829676i	\(0.311475\pi\)
\(13\)	15.7679	+	15.7679i	1.21292	+	1.21292i	0.970062	+	0.242855i	\(0.0780841\pi\)
							0.242855	+	0.970062i	\(0.421916\pi\)
\(17\)	−12.8903	−	7.44223i	−0.758254	−	0.437778i	0.0704142	−	0.997518i	\(-0.477568\pi\)
							−0.828669	+	0.559739i	\(0.810901\pi\)
\(19\)	8.66278	−	32.3299i	0.455936	−	1.70158i	−0.229386	−	0.973336i	\(-0.573672\pi\)
							0.685322	−	0.728241i	\(-0.259662\pi\)
\(23\)	18.7330	−	10.8155i	0.814479	−	0.470240i	−0.0340296	−	0.999421i	\(-0.510834\pi\)
							0.848509	+	0.529181i	\(0.177501\pi\)
\(29\)	−7.54368	+	7.54368i	−0.260127	+	0.260127i	−0.825106	−	0.564979i	\(-0.808884\pi\)
							0.564979	+	0.825106i	\(0.308884\pi\)
\(31\)	34.0116	+	19.6366i	1.09715	+	0.633440i	0.935471	−	0.353404i	\(-0.114976\pi\)
							0.161679	+	0.986843i	\(0.448309\pi\)
\(37\)	5.60270	−	20.9095i	0.151424	−	0.565123i	−0.847961	−	0.530059i	\(-0.822170\pi\)
							0.999385	−	0.0350640i	\(-0.0111635\pi\)
\(41\)	53.6306			1.30806			0.654031	−	0.756467i	\(-0.273076\pi\)
							0.654031	+	0.756467i	\(0.273076\pi\)
\(43\)	−26.3556	−	26.3556i	−0.612920	−	0.612920i	0.330786	−	0.943706i	\(-0.392686\pi\)
							−0.943706	+	0.330786i	\(0.892686\pi\)
\(47\)	4.19507	−	2.42203i	0.0892569	−	0.0515325i	−0.454707	−	0.890641i	\(-0.650256\pi\)
							0.543964	+	0.839108i	\(0.316923\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.48	yes	256
7.5	odd	6	inner	336.3.bv.a.229.6	yes	256
16.13	even	4	inner	336.3.bv.a.157.6	yes	256
112.61	odd	12	inner	336.3.bv.a.61.48	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.48	✓	256	112.61	odd	12	inner
336.3.bv.a.157.6	yes	256	16.13	even	4	inner
336.3.bv.a.229.6	yes	256	7.5	odd	6	inner
336.3.bv.a.325.48	yes	256	1.1	even	1	trivial