Embedded newform 336.3.bv.a.325.3

• Label: 336.3.bv.a.325.3
• 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			325.3
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99417 - 0.152626i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(3.95341 + 0.608722i) q^{4} +(-2.28005 - 0.610938i) q^{5} +(3.40473 - 0.638613i) q^{6} +(3.56751 - 6.02269i) q^{7} +(-7.79086 - 1.81729i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99417	−	0.152626i	−0.997084	−	0.0763128i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−2.28005	−	0.610938i	−0.456011	−	0.122188i								0.0234998	−	0.999724i	\(-0.492519\pi\)
							−0.479510	+	0.877536i	\(0.659186\pi\)
\(7\)	3.56751	−	6.02269i	0.509645	−	0.860385i
							\(11\)	−6.72682	+	1.80245i	−0.611529	+	0.163859i	−0.551274	−	0.834324i	\(-0.685858\pi\)
−0.0602552	+	0.998183i	\(0.519191\pi\)
\(13\)	7.56701	+	7.56701i	0.582078	+	0.582078i	0.935474	−	0.353396i	\(-0.114973\pi\)
							−0.353396	+	0.935474i	\(0.614973\pi\)
\(17\)	−3.20303	−	1.84927i	−0.188413	−	0.108780i	0.402826	−	0.915276i	\(-0.368028\pi\)
							−0.591240	+	0.806496i	\(0.701361\pi\)
\(19\)	−0.746430	+	2.78571i	−0.0392858	+	0.146616i	−0.982783	−	0.184764i	\(-0.940848\pi\)
							0.943497	+	0.331381i	\(0.107514\pi\)
\(23\)	16.2746	−	9.39614i	0.707591	−	0.408528i	−0.102578	−	0.994725i	\(-0.532709\pi\)
							0.810168	+	0.586197i	\(0.199376\pi\)
\(29\)	−13.6896	+	13.6896i	−0.472056	+	0.472056i	−0.902579	−	0.430523i	\(-0.858329\pi\)
							0.430523	+	0.902579i	\(0.358329\pi\)
\(31\)	−35.3189	−	20.3914i	−1.13932	−	0.657787i	−0.193058	−	0.981187i	\(-0.561841\pi\)
							−0.946262	+	0.323400i	\(0.895174\pi\)
\(37\)	−0.0747757	+	0.279067i	−0.00202096	+	0.00754234i	−0.966929	−	0.255046i	\(-0.917909\pi\)
							0.964908	+	0.262588i	\(0.0845761\pi\)
\(41\)	−55.9688			−1.36509			−0.682547	−	0.730842i	\(-0.739128\pi\)
							−0.682547	+	0.730842i	\(0.739128\pi\)
\(43\)	−9.17097	−	9.17097i	−0.213278	−	0.213278i	0.592380	−	0.805659i	\(-0.298188\pi\)
							−0.805659	+	0.592380i	\(0.798188\pi\)
\(47\)	−32.1086	+	18.5379i	−0.683161	+	0.394423i	−0.801045	−	0.598604i	\(-0.795722\pi\)
							0.117884	+	0.993027i	\(0.462389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.3	yes	256
7.5	odd	6	inner	336.3.bv.a.229.42	yes	256
16.13	even	4	inner	336.3.bv.a.157.42	yes	256
112.61	odd	12	inner	336.3.bv.a.61.3	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.3	✓	256	112.61	odd	12	inner
336.3.bv.a.157.42	yes	256	16.13	even	4	inner
336.3.bv.a.229.42	yes	256	7.5	odd	6	inner
336.3.bv.a.325.3	yes	256	1.1	even	1	trivial