Embedded newform 336.3.bv.a.325.61

• Label: 336.3.bv.a.325.61
• 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			325.61
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.97626 - 0.307219i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(3.81123 - 1.21429i) q^{4} +(-0.857813 - 0.229850i) q^{5} +(-3.16863 + 1.39992i) q^{6} +(-2.07138 - 6.68651i) q^{7} +(7.15895 - 3.57064i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.97626	−	0.307219i	0.988132	−	0.153610i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−0.857813	−	0.229850i	−0.171563	−	0.0459701i								0.172015	−	0.985094i	\(-0.444972\pi\)
							−0.343578	+	0.939124i	\(0.611639\pi\)
\(7\)	−2.07138	−	6.68651i	−0.295912	−	0.955215i
							\(11\)	8.56527	−	2.29506i	0.778661	−	0.208642i	0.152467	−	0.988309i	\(-0.451278\pi\)
0.626194	+	0.779667i	\(0.284612\pi\)
\(13\)	−3.60255	−	3.60255i	−0.277119	−	0.277119i	0.554839	−	0.831958i	\(-0.312780\pi\)
							−0.831958	+	0.554839i	\(0.812780\pi\)
\(17\)	−5.94423	−	3.43190i	−0.349661	−	0.201877i	0.314875	−	0.949133i	\(-0.398037\pi\)
							−0.664536	+	0.747256i	\(0.731371\pi\)
\(19\)	3.67384	−	13.7110i	0.193360	−	0.721630i	−0.799325	−	0.600899i	\(-0.794809\pi\)
							0.992685	−	0.120731i	\(-0.0385239\pi\)
\(23\)	15.2019	−	8.77683i	0.660953	−	0.381601i	−0.131687	−	0.991291i	\(-0.542039\pi\)
							0.792640	+	0.609690i	\(0.208706\pi\)
\(29\)	32.4748	−	32.4748i	1.11982	−	1.11982i	0.128052	−	0.991767i	\(-0.459128\pi\)
							0.991767	−	0.128052i	\(-0.0408725\pi\)
\(31\)	37.9077	+	21.8860i	1.22283	+	0.706000i	0.965520	−	0.260329i	\(-0.0838310\pi\)
							0.257308	+	0.966329i	\(0.417164\pi\)
\(37\)	−12.7528	+	47.5940i	−0.344670	+	1.28633i	0.548328	+	0.836264i	\(0.315265\pi\)
							−0.892997	+	0.450062i	\(0.851402\pi\)
\(41\)	−29.4677			−0.718725			−0.359362	−	0.933198i	\(-0.617006\pi\)
							−0.359362	+	0.933198i	\(0.617006\pi\)
\(43\)	23.0606	+	23.0606i	0.536292	+	0.536292i	0.922438	−	0.386146i	\(-0.126194\pi\)
							−0.386146	+	0.922438i	\(0.626194\pi\)
\(47\)	−37.2089	+	21.4826i	−0.791678	+	0.457076i	−0.840553	−	0.541729i	\(-0.817770\pi\)
							0.0488748	+	0.998805i	\(0.484436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.61	yes	256
7.5	odd	6	inner	336.3.bv.a.229.19	yes	256
16.13	even	4	inner	336.3.bv.a.157.19	yes	256
112.61	odd	12	inner	336.3.bv.a.61.61	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.61	✓	256	112.61	odd	12	inner
336.3.bv.a.157.19	yes	256	16.13	even	4	inner
336.3.bv.a.229.19	yes	256	7.5	odd	6	inner
336.3.bv.a.325.61	yes	256	1.1	even	1	trivial