Embedded newform 336.2.bs.a.19.19

• Label: 336.2.bs.a.19.19
• Level: $336$
• Weight: $2$
• Character: 336.19
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.19
Character	\(\chi\)	\(=\)	336.19
Dual form			336.2.bs.a.283.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.278250 - 1.38657i) q^{2} +(-0.258819 + 0.965926i) q^{3} +(-1.84515 - 0.771627i) q^{4} +(-1.22597 + 0.328499i) q^{5} +(1.26731 + 0.627640i) q^{6} +(-2.24886 - 1.39379i) q^{7} +(-1.58333 + 2.34373i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 4 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.278250	−	1.38657i	0.196753	−	0.980453i
							\(3\)	−0.258819	+	0.965926i	−0.149429	+	0.557678i
\(5\)	−1.22597	+	0.328499i	−0.548272	+	0.146909i								−0.522313	−	0.852754i	\(-0.674931\pi\)
							−0.0259594	+	0.999663i	\(0.508264\pi\)
\(7\)	−2.24886	−	1.39379i	−0.849988	−	0.526801i
							\(11\)	0.145015	−	0.541203i	0.0437236	−	0.163179i	−0.940612	−	0.339483i	\(-0.889748\pi\)
0.984336	+	0.176305i	\(0.0564144\pi\)
\(13\)	−2.12063	+	2.12063i	−0.588156	+	0.588156i	−0.937132	−	0.348976i	\(-0.886530\pi\)
							0.348976	+	0.937132i	\(0.386530\pi\)
\(17\)	−4.20834	+	2.42969i	−1.02067	+	0.589286i	−0.914299	−	0.405040i	\(-0.867258\pi\)
							−0.106374	+	0.994326i	\(0.533924\pi\)
\(19\)	−6.59718	+	1.76771i	−1.51350	+	0.405540i	−0.917595	−	0.397518i	\(-0.869872\pi\)
							−0.595901	+	0.803058i	\(0.703205\pi\)
\(23\)	2.91674	−	5.05194i	0.608182	−	1.05340i	−0.383358	−	0.923600i	\(-0.625233\pi\)
							0.991540	−	0.129802i	\(-0.0414341\pi\)
\(29\)	3.83987	+	3.83987i	0.713046	+	0.713046i	0.967171	−	0.254125i	\(-0.0817875\pi\)
							−0.254125	+	0.967171i	\(0.581788\pi\)
\(31\)	−3.58919	−	6.21665i	−0.644637	−	1.11654i	−0.984385	−	0.176027i	\(-0.943675\pi\)
							0.339748	−	0.940516i	\(-0.389658\pi\)
\(37\)	−2.32272	−	8.66852i	−0.381853	−	1.42510i	−0.843068	−	0.537807i	\(-0.819253\pi\)
							0.461215	−	0.887288i	\(-0.347414\pi\)
\(41\)	2.07800			0.324529			0.162265	−	0.986747i	\(-0.448120\pi\)
							0.162265	+	0.986747i	\(0.448120\pi\)
\(43\)	1.61407	+	1.61407i	0.246143	+	0.246143i	0.819386	−	0.573242i	\(-0.194315\pi\)
							−0.573242	+	0.819386i	\(0.694315\pi\)
\(47\)	3.85125	−	6.67057i	0.561763	−	0.973002i	−0.435580	−	0.900150i	\(-0.643457\pi\)
							0.997343	−	0.0728520i	\(-0.0232101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bs.a.19.19	✓	128
7.3	odd	6	inner	336.2.bs.a.115.25	yes	128
16.11	odd	4	inner	336.2.bs.a.187.25	yes	128
112.59	even	12	inner	336.2.bs.a.283.19	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bs.a.19.19	✓	128	1.1	even	1	trivial
336.2.bs.a.115.25	yes	128	7.3	odd	6	inner
336.2.bs.a.187.25	yes	128	16.11	odd	4	inner
336.2.bs.a.283.19	yes	128	112.59	even	12	inner