Embedded newform 336.3.bv.a.325.64

• Label: 336.3.bv.a.325.64
• 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			325.64
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.99987 + 0.0230213i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(3.99894 + 0.0920790i) q^{4} +(9.11045 + 2.44114i) q^{5} +(-3.35616 + 0.858001i) q^{6} +(-1.70773 + 6.78850i) q^{7} +(7.99523 + 0.276207i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.99987	+	0.0230213i	0.999934	+	0.0115106i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	9.11045	+	2.44114i	1.82209	+	0.488228i								0.997044	−	0.0768361i	\(-0.0244818\pi\)
							0.825047	+	0.565064i	\(0.191148\pi\)
\(7\)	−1.70773	+	6.78850i	−0.243961	+	0.969785i
							\(11\)	−6.13208	+	1.64309i	−0.557462	+	0.149371i	−0.526540	−	0.850150i	\(-0.676511\pi\)
−0.0309218	+	0.999522i	\(0.509844\pi\)
\(13\)	−4.00215	−	4.00215i	−0.307858	−	0.307858i	0.536220	−	0.844078i	\(-0.319852\pi\)
							−0.844078	+	0.536220i	\(0.819852\pi\)
\(17\)	−18.1883	−	10.5010i	−1.06990	−	0.617708i	−0.141747	−	0.989903i	\(-0.545272\pi\)
							−0.928155	+	0.372195i	\(0.878605\pi\)
\(19\)	−1.99384	+	7.44113i	−0.104939	+	0.391638i	−0.998338	−	0.0576256i	\(-0.981647\pi\)
							0.893399	+	0.449264i	\(0.148314\pi\)
\(23\)	−7.77981	+	4.49167i	−0.338253	+	0.195290i	−0.659499	−	0.751705i	\(-0.729232\pi\)
							0.321246	+	0.946996i	\(0.395898\pi\)
\(29\)	31.8628	−	31.8628i	1.09872	−	1.09872i	0.104158	−	0.994561i	\(-0.466785\pi\)
							0.994561	−	0.104158i	\(-0.0332147\pi\)
\(31\)	32.4953	+	18.7612i	1.04824	+	0.605199i	0.922155	−	0.386821i	\(-0.126427\pi\)
							0.126081	+	0.992020i	\(0.459760\pi\)
\(37\)	10.5717	−	39.4541i	0.285722	−	1.06633i	−0.662589	−	0.748983i	\(-0.730542\pi\)
							0.948310	−	0.317344i	\(-0.102791\pi\)
\(41\)	−42.0874			−1.02652			−0.513261	−	0.858233i	\(-0.671563\pi\)
							−0.513261	+	0.858233i	\(0.671563\pi\)
\(43\)	−45.6954	−	45.6954i	−1.06268	−	1.06268i	−0.997899	−	0.0647839i	\(-0.979364\pi\)
							−0.0647839	−	0.997899i	\(-0.520636\pi\)
\(47\)	40.3211	−	23.2794i	0.857896	−	0.495306i	−0.00541127	−	0.999985i	\(-0.501722\pi\)
							0.863307	+	0.504679i	\(0.168389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.64	yes	256
7.5	odd	6	inner	336.3.bv.a.229.23	yes	256
16.13	even	4	inner	336.3.bv.a.157.23	yes	256
112.61	odd	12	inner	336.3.bv.a.61.64	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.64	✓	256	112.61	odd	12	inner
336.3.bv.a.157.23	yes	256	16.13	even	4	inner
336.3.bv.a.229.23	yes	256	7.5	odd	6	inner
336.3.bv.a.325.64	yes	256	1.1	even	1	trivial