Embedded newform 336.3.bv.a.325.1

• Label: 336.3.bv.a.325.1
• 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			325.1
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00000 - 0.00346047i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(3.99998 + 0.0138419i) q^{4} +(0.893057 + 0.239294i) q^{5} +(3.34761 - 0.890785i) q^{6} +(-6.85042 + 1.43938i) q^{7} +(-7.99989 - 0.0415255i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−	0.00346047i	−0.999999	−	0.00173023i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	0.893057	+	0.239294i	0.178611	+	0.0478588i								0.347016	−	0.937859i	\(-0.387195\pi\)
							−0.168405	+	0.985718i	\(0.553862\pi\)
\(7\)	−6.85042	+	1.43938i	−0.978631	+	0.205626i
							\(11\)	13.1400	−	3.52087i	1.19455	−	0.320079i	0.393867	−	0.919167i	\(-0.371137\pi\)
0.800683	+	0.599089i	\(0.204470\pi\)
\(13\)	−5.95384	−	5.95384i	−0.457988	−	0.457988i	0.440007	−	0.897994i	\(-0.354976\pi\)
							−0.897994	+	0.440007i	\(0.854976\pi\)
\(17\)	−7.06773	−	4.08056i	−0.415749	−	0.240033i	0.277508	−	0.960723i	\(-0.410492\pi\)
							−0.693257	+	0.720691i	\(0.743825\pi\)
\(19\)	−6.93025	+	25.8641i	−0.364750	+	1.36127i	0.503009	+	0.864281i	\(0.332226\pi\)
							−0.867759	+	0.496985i	\(0.834440\pi\)
\(23\)	17.5692	−	10.1436i	0.763877	−	0.441024i	−0.0668093	−	0.997766i	\(-0.521282\pi\)
							0.830686	+	0.556741i	\(0.187949\pi\)
\(29\)	23.5077	−	23.5077i	0.810611	−	0.810611i	−0.174114	−	0.984725i	\(-0.555706\pi\)
							0.984725	+	0.174114i	\(0.0557062\pi\)
\(31\)	22.8280	+	13.1798i	0.736387	+	0.425153i	0.820754	−	0.571281i	\(-0.193554\pi\)
							−0.0843671	+	0.996435i	\(0.526887\pi\)
\(37\)	10.0783	−	37.6127i	0.272386	−	1.01656i	−0.685187	−	0.728367i	\(-0.740279\pi\)
							0.957573	−	0.288191i	\(-0.0930539\pi\)
\(41\)	69.5815			1.69711			0.848555	−	0.529106i	\(-0.177473\pi\)
							0.848555	+	0.529106i	\(0.177473\pi\)
\(43\)	49.4595	+	49.4595i	1.15022	+	1.15022i	0.986508	+	0.163713i	\(0.0523472\pi\)
							0.163713	+	0.986508i	\(0.447653\pi\)
\(47\)	64.1983	−	37.0649i	1.36592	−	0.788615i	0.375517	−	0.926815i	\(-0.377465\pi\)
							0.990404	+	0.138200i	\(0.0441318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.1	yes	256
7.5	odd	6	inner	336.3.bv.a.229.44	yes	256
16.13	even	4	inner	336.3.bv.a.157.44	yes	256
112.61	odd	12	inner	336.3.bv.a.61.1	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.1	✓	256	112.61	odd	12	inner
336.3.bv.a.157.44	yes	256	16.13	even	4	inner
336.3.bv.a.229.44	yes	256	7.5	odd	6	inner
336.3.bv.a.325.1	yes	256	1.1	even	1	trivial