Embedded newform 336.3.bv.a.325.8

• Label: 336.3.bv.a.325.8
• 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			325.8
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81659 + 0.836654i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(2.60002 - 3.03972i) q^{4} +(-7.08579 - 1.89863i) q^{5} +(2.66416 - 2.21411i) q^{6} +(3.71636 + 5.93201i) q^{7} +(-2.17999 + 7.69725i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81659	+	0.836654i	−0.908297	+	0.418327i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−7.08579	−	1.89863i	−1.41716	−	0.379727i								−0.532684	−	0.846314i	\(-0.678817\pi\)
							−0.884475	+	0.466588i	\(0.845483\pi\)
\(7\)	3.71636	+	5.93201i	0.530908	+	0.847429i
							\(11\)	17.9917	−	4.82086i	1.63561	−	0.438260i	0.680075	−	0.733142i	\(-0.261947\pi\)
0.955534	+	0.294882i	\(0.0952803\pi\)
\(13\)	7.01233	+	7.01233i	0.539410	+	0.539410i	0.923356	−	0.383946i	\(-0.125435\pi\)
							−0.383946	+	0.923356i	\(0.625435\pi\)
\(17\)	−20.0234	−	11.5605i	−1.17785	−	0.680032i	−0.222334	−	0.974971i	\(-0.571367\pi\)
							−0.955516	+	0.294939i	\(0.904701\pi\)
\(19\)	−0.314773	+	1.17475i	−0.0165670	+	0.0618290i	−0.973714	−	0.227772i	\(-0.926856\pi\)
							0.957147	+	0.289601i	\(0.0935226\pi\)
\(23\)	−29.0338	+	16.7627i	−1.26234	+	0.728813i	−0.973527	−	0.228573i	\(-0.926594\pi\)
							−0.288814	+	0.957385i	\(0.593261\pi\)
\(29\)	3.41534	−	3.41534i	0.117770	−	0.117770i	−0.645766	−	0.763536i	\(-0.723462\pi\)
							0.763536	+	0.645766i	\(0.223462\pi\)
\(31\)	−15.4961	−	8.94665i	−0.499873	−	0.288602i	0.228788	−	0.973476i	\(-0.426524\pi\)
							−0.728661	+	0.684875i	\(0.759857\pi\)
\(37\)	−14.7092	+	54.8955i	−0.397546	+	1.48366i	0.419853	+	0.907592i	\(0.362081\pi\)
							−0.817400	+	0.576071i	\(0.804585\pi\)
\(41\)	−64.0845			−1.56304			−0.781518	−	0.623882i	\(-0.785554\pi\)
							−0.781518	+	0.623882i	\(0.785554\pi\)
\(43\)	−45.2115	−	45.2115i	−1.05143	−	1.05143i	−0.998604	−	0.0528265i	\(-0.983177\pi\)
							−0.0528265	−	0.998604i	\(-0.516823\pi\)
\(47\)	−28.0819	+	16.2131i	−0.597488	+	0.344960i	−0.768053	−	0.640387i	\(-0.778774\pi\)
							0.170565	+	0.985346i	\(0.445441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.8	yes	256
7.5	odd	6	inner	336.3.bv.a.229.53	yes	256
16.13	even	4	inner	336.3.bv.a.157.53	yes	256
112.61	odd	12	inner	336.3.bv.a.61.8	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.8	✓	256	112.61	odd	12	inner
336.3.bv.a.157.53	yes	256	16.13	even	4	inner
336.3.bv.a.229.53	yes	256	7.5	odd	6	inner
336.3.bv.a.325.8	yes	256	1.1	even	1	trivial