Embedded newform 336.3.bv.a.325.24

• Label: 336.3.bv.a.325.24
• 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			325.24
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.702257 + 1.87265i) q^{2} +(1.67303 - 0.448288i) q^{3} +(-3.01367 - 2.63017i) q^{4} +(6.23586 + 1.67089i) q^{5} +(-0.335411 + 3.44783i) q^{6} +(-4.88285 - 5.01576i) q^{7} +(7.04177 - 3.79651i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.702257	+	1.87265i	−0.351128	+	0.936327i
							\(3\)	1.67303	−	0.448288i	0.557678	−	0.149429i
\(5\)	6.23586	+	1.67089i	1.24717	+	0.334179i								0.821243	−	0.570578i	\(-0.193281\pi\)
							0.425929	+	0.904757i	\(0.359947\pi\)
\(7\)	−4.88285	−	5.01576i	−0.697550	−	0.716537i
							\(11\)	9.22710	−	2.47239i	0.838827	−	0.224763i	0.186266	−	0.982499i	\(-0.440361\pi\)
0.652561	+	0.757736i	\(0.273695\pi\)
\(13\)	−2.35411	−	2.35411i	−0.181086	−	0.181086i	0.610743	−	0.791829i	\(-0.290871\pi\)
							−0.791829	+	0.610743i	\(0.790871\pi\)
\(17\)	−0.189303	−	0.109294i	−0.0111355	−	0.00642907i	0.494422	−	0.869222i	\(-0.335380\pi\)
							−0.505557	+	0.862793i	\(0.668713\pi\)
\(19\)	5.77149	−	21.5395i	0.303763	−	1.13366i	−0.630242	−	0.776399i	\(-0.717044\pi\)
							0.934005	−	0.357260i	\(-0.116289\pi\)
\(23\)	31.5046	−	18.1892i	1.36977	−	0.790835i	0.378869	−	0.925450i	\(-0.376313\pi\)
							0.990898	+	0.134615i	\(0.0429798\pi\)
\(29\)	−25.7335	+	25.7335i	−0.887362	+	0.887362i	−0.994269	−	0.106907i	\(-0.965905\pi\)
							0.106907	+	0.994269i	\(0.465905\pi\)
\(31\)	32.8537	+	18.9681i	1.05980	+	0.611873i	0.925376	−	0.379051i	\(-0.123750\pi\)
							0.134420	+	0.990924i	\(0.457083\pi\)
\(37\)	5.78897	−	21.6047i	0.156459	−	0.583912i	−0.842517	−	0.538669i	\(-0.818927\pi\)
							0.998976	−	0.0452424i	\(-0.0144060\pi\)
\(41\)	45.8489			1.11827			0.559134	−	0.829078i	\(-0.311134\pi\)
							0.559134	+	0.829078i	\(0.311134\pi\)
\(43\)	11.7755	+	11.7755i	0.273850	+	0.273850i	0.830648	−	0.556798i	\(-0.187970\pi\)
							−0.556798	+	0.830648i	\(0.687970\pi\)
\(47\)	−18.8256	+	10.8690i	−0.400545	+	0.231254i	−0.686719	−	0.726923i	\(-0.740950\pi\)
							0.286174	+	0.958178i	\(0.407616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.24	yes	256
7.5	odd	6	inner	336.3.bv.a.229.62	yes	256
16.13	even	4	inner	336.3.bv.a.157.62	yes	256
112.61	odd	12	inner	336.3.bv.a.61.24	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.24	✓	256	112.61	odd	12	inner
336.3.bv.a.157.62	yes	256	16.13	even	4	inner
336.3.bv.a.229.62	yes	256	7.5	odd	6	inner
336.3.bv.a.325.24	yes	256	1.1	even	1	trivial