Embedded newform 336.3.bv.a.325.17

• Label: 336.3.bv.a.325.17
• 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			325.17
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35433 + 1.47166i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-0.331561 - 3.98623i) q^{4} +(-2.13659 - 0.572498i) q^{5} +(-1.60612 + 3.06926i) q^{6} +(0.973371 - 6.93199i) q^{7} +(6.31542 + 4.91075i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35433	+	1.47166i	−0.677167	+	0.735830i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	−2.13659	−	0.572498i	−0.427318	−	0.114500i								0.0387487	−	0.999249i	\(-0.487663\pi\)
							−0.466067	+	0.884749i	\(0.654329\pi\)
\(7\)	0.973371	−	6.93199i	0.139053	−	0.990285i
							\(11\)	−13.3375	+	3.57377i	−1.21250	+	0.324888i	−0.807742	−	0.589536i	\(-0.799310\pi\)
−0.404757	+	0.914424i	\(0.632644\pi\)
\(13\)	2.55547	+	2.55547i	0.196574	+	0.196574i	0.798530	−	0.601955i	\(-0.205611\pi\)
							−0.601955	+	0.798530i	\(0.705611\pi\)
\(17\)	−12.7638	−	7.36919i	−0.750813	−	0.433482i	0.0751748	−	0.997170i	\(-0.476049\pi\)
							−0.825987	+	0.563688i	\(0.809382\pi\)
\(19\)	1.95144	−	7.28289i	0.102708	−	0.383310i	−0.895367	−	0.445328i	\(-0.853087\pi\)
							0.998075	+	0.0620183i	\(0.0197537\pi\)
\(23\)	−27.0751	+	15.6318i	−1.17718	+	0.679643i	−0.955360	−	0.295445i	\(-0.904532\pi\)
							−0.221817	+	0.975088i	\(0.571199\pi\)
\(29\)	5.66444	−	5.66444i	0.195326	−	0.195326i	−0.602667	−	0.797993i	\(-0.705895\pi\)
							0.797993	+	0.602667i	\(0.205895\pi\)
\(31\)	−24.6045	−	14.2054i	−0.793694	−	0.458239i	0.0475675	−	0.998868i	\(-0.484853\pi\)
							−0.841261	+	0.540629i	\(0.818186\pi\)
\(37\)	1.40112	−	5.22906i	0.0378682	−	0.141326i	−0.944404	−	0.328789i	\(-0.893360\pi\)
							0.982272	+	0.187463i	\(0.0600263\pi\)
\(41\)	−24.1419			−0.588827			−0.294413	−	0.955678i	\(-0.595124\pi\)
							−0.294413	+	0.955678i	\(0.595124\pi\)
\(43\)	2.97874	+	2.97874i	0.0692729	+	0.0692729i	0.740894	−	0.671621i	\(-0.234402\pi\)
							−0.671621	+	0.740894i	\(0.734402\pi\)
\(47\)	20.0956	−	11.6022i	0.427567	−	0.246856i	−0.270743	−	0.962652i	\(-0.587269\pi\)
							0.698309	+	0.715796i	\(0.253936\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.17	yes	256
7.5	odd	6	inner	336.3.bv.a.229.60	yes	256
16.13	even	4	inner	336.3.bv.a.157.60	yes	256
112.61	odd	12	inner	336.3.bv.a.61.17	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.17	✓	256	112.61	odd	12	inner
336.3.bv.a.157.60	yes	256	16.13	even	4	inner
336.3.bv.a.229.60	yes	256	7.5	odd	6	inner
336.3.bv.a.325.17	yes	256	1.1	even	1	trivial