Embedded newform 336.3.bv.a.325.22

• Label: 336.3.bv.a.325.22
• 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			325.22
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.804536 - 1.83104i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(-2.70544 + 2.94628i) q^{4} +(2.18224 + 0.584730i) q^{5} +(2.16685 + 2.70273i) q^{6} +(0.147162 + 6.99845i) q^{7} +(7.57140 + 2.58339i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.804536	−	1.83104i	−0.402268	−	0.915522i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	2.18224	+	0.584730i	0.436449	+	0.116946i								0.470352	−	0.882479i	\(-0.344127\pi\)
							−0.0339030	+	0.999425i	\(0.510794\pi\)
\(7\)	0.147162	+	6.99845i	0.0210231	+	0.999779i
							\(11\)	−2.12655	+	0.569807i	−0.193322	+	0.0518006i	−0.354181	−	0.935177i	\(-0.615240\pi\)
0.160859	+	0.986977i	\(0.448574\pi\)
\(13\)	−15.0388	−	15.0388i	−1.15683	−	1.15683i	−0.985154	−	0.171676i	\(-0.945082\pi\)
							−0.171676	−	0.985154i	\(-0.554918\pi\)
\(17\)	−10.9817	−	6.34027i	−0.645981	−	0.372957i	0.140934	−	0.990019i	\(-0.454990\pi\)
							−0.786915	+	0.617062i	\(0.788323\pi\)
\(19\)	−1.15813	+	4.32222i	−0.0609544	+	0.227485i	−0.989683	−	0.143276i	\(-0.954236\pi\)
							0.928728	+	0.370761i	\(0.120903\pi\)
\(23\)	21.6623	−	12.5068i	0.941841	−	0.543772i	0.0513038	−	0.998683i	\(-0.483662\pi\)
							0.890537	+	0.454911i	\(0.150329\pi\)
\(29\)	−38.5944	+	38.5944i	−1.33084	+	1.33084i	−0.426222	+	0.904619i	\(0.640156\pi\)
							−0.904619	+	0.426222i	\(0.859844\pi\)
\(31\)	−5.51120	−	3.18189i	−0.177781	−	0.102642i	0.408469	−	0.912772i	\(-0.366063\pi\)
							−0.586250	+	0.810131i	\(0.699396\pi\)
\(37\)	1.28233	−	4.78572i	0.0346576	−	0.129344i	−0.946430	−	0.322910i	\(-0.895339\pi\)
							0.981087	+	0.193566i	\(0.0620055\pi\)
\(41\)	−3.99789			−0.0975095			−0.0487548	−	0.998811i	\(-0.515525\pi\)
							−0.0487548	+	0.998811i	\(0.515525\pi\)
\(43\)	−30.3342	−	30.3342i	−0.705446	−	0.705446i	0.260128	−	0.965574i	\(-0.416235\pi\)
							−0.965574	+	0.260128i	\(0.916235\pi\)
\(47\)	−10.4939	+	6.05867i	−0.223275	+	0.128908i	−0.607466	−	0.794346i	\(-0.707814\pi\)
							0.384191	+	0.923254i	\(0.374480\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.22	yes	256
7.5	odd	6	inner	336.3.bv.a.229.21	yes	256
16.13	even	4	inner	336.3.bv.a.157.21	yes	256
112.61	odd	12	inner	336.3.bv.a.61.22	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.22	✓	256	112.61	odd	12	inner
336.3.bv.a.157.21	yes	256	16.13	even	4	inner
336.3.bv.a.229.21	yes	256	7.5	odd	6	inner
336.3.bv.a.325.22	yes	256	1.1	even	1	trivial