Embedded newform 336.3.bv.a.325.27

• Label: 336.3.bv.a.325.27
• 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			325.27
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.477095 - 1.94226i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-3.54476 + 1.85329i) q^{4} +(-8.55653 - 2.29272i) q^{5} +(-1.66889 - 3.03559i) q^{6} +(3.91477 - 5.80298i) q^{7} +(5.29075 + 6.00066i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.477095	−	1.94226i	−0.238547	−	0.971131i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−8.55653	−	2.29272i	−1.71131	−	0.458543i								−0.735562	−	0.677457i	\(-0.763082\pi\)
							−0.975744	+	0.218914i	\(0.929749\pi\)
\(7\)	3.91477	−	5.80298i	0.559253	−	0.828997i
							\(11\)	−9.05424	+	2.42608i	−0.823113	+	0.220552i	−0.645707	−	0.763585i	\(-0.723437\pi\)
−0.177406	+	0.984138i	\(0.556771\pi\)
\(13\)	−5.04174	−	5.04174i	−0.387826	−	0.387826i	0.486085	−	0.873912i	\(-0.338425\pi\)
							−0.873912	+	0.486085i	\(0.838425\pi\)
\(17\)	11.7847	+	6.80389i	0.693217	+	0.400229i	0.804816	−	0.593525i	\(-0.202264\pi\)
							−0.111599	+	0.993753i	\(0.535597\pi\)
\(19\)	−8.36309	+	31.2115i	−0.440163	+	1.64271i	0.288238	+	0.957559i	\(0.406931\pi\)
							−0.728401	+	0.685151i	\(0.759736\pi\)
\(23\)	−27.6469	+	15.9620i	−1.20204	+	0.693998i	−0.961008	−	0.276520i	\(-0.910819\pi\)
							−0.241031	+	0.970517i	\(0.577486\pi\)
\(29\)	−19.5228	+	19.5228i	−0.673200	+	0.673200i	−0.958452	−	0.285253i	\(-0.907922\pi\)
							0.285253	+	0.958452i	\(0.407922\pi\)
\(31\)	24.2582	+	14.0055i	0.782522	+	0.451789i	0.837323	−	0.546708i	\(-0.184119\pi\)
							−0.0548014	+	0.998497i	\(0.517453\pi\)
\(37\)	1.92831	−	7.19656i	0.0521166	−	0.194502i	−0.934959	−	0.354755i	\(-0.884564\pi\)
							0.987076	+	0.160253i	\(0.0512310\pi\)
\(41\)	−24.7338			−0.603262			−0.301631	−	0.953425i	\(-0.597531\pi\)
							−0.301631	+	0.953425i	\(0.597531\pi\)
\(43\)	−35.1088	−	35.1088i	−0.816483	−	0.816483i	0.169114	−	0.985597i	\(-0.445909\pi\)
							−0.985597	+	0.169114i	\(0.945909\pi\)
\(47\)	7.84920	−	4.53174i	0.167004	−	0.0964200i	−0.414168	−	0.910200i	\(-0.635927\pi\)
							0.581173	+	0.813780i	\(0.302594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.27	yes	256
7.5	odd	6	inner	336.3.bv.a.229.16	yes	256
16.13	even	4	inner	336.3.bv.a.157.16	yes	256
112.61	odd	12	inner	336.3.bv.a.61.27	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.27	✓	256	112.61	odd	12	inner
336.3.bv.a.157.16	yes	256	16.13	even	4	inner
336.3.bv.a.229.16	yes	256	7.5	odd	6	inner
336.3.bv.a.325.27	yes	256	1.1	even	1	trivial