Embedded newform 336.2.bs.a.19.16

• Label: 336.2.bs.a.19.16
• 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			19.16
Character	\(\chi\)	\(=\)	336.19
Dual form			336.2.bs.a.283.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0555893 + 1.41312i) q^{2} +(0.258819 - 0.965926i) q^{3} +(-1.99382 + 0.157109i) q^{4} +(-2.15146 + 0.576482i) q^{5} +(1.37936 + 0.312047i) q^{6} +(0.0219879 - 2.64566i) q^{7} +(-0.332849 - 2.80877i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 4 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0555893	+	1.41312i	0.0393076	+	0.999227i
							\(3\)	0.258819	−	0.965926i	0.149429	−	0.557678i
\(5\)	−2.15146	+	0.576482i	−0.962162	+	0.257810i								−0.705515	−	0.708695i	\(-0.749284\pi\)
							−0.256647	+	0.966505i	\(0.582618\pi\)
\(7\)	0.0219879	−	2.64566i	0.00831066	−	0.999965i
							\(11\)	0.760988	−	2.84005i	0.229447	−	0.856306i	−0.751128	−	0.660157i	\(-0.770490\pi\)
0.980574	−	0.196149i	\(-0.0628436\pi\)
\(13\)	3.96933	−	3.96933i	1.10090	−	1.10090i	0.106593	−	0.994303i	\(-0.466006\pi\)
							0.994303	−	0.106593i	\(-0.0339941\pi\)
\(17\)	−4.60639	+	2.65950i	−1.11721	+	0.645023i	−0.940688	−	0.339273i	\(-0.889819\pi\)
							−0.176525	+	0.984296i	\(0.556486\pi\)
\(19\)	−0.0198339	+	0.00531448i	−0.00455021	+	0.00121922i	−0.261093	−	0.965314i	\(-0.584083\pi\)
							0.256543	+	0.966533i	\(0.417416\pi\)
\(23\)	2.18808	−	3.78987i	0.456246	−	0.790242i	−0.542513	−	0.840048i	\(-0.682527\pi\)
							0.998759	+	0.0498058i	\(0.0158602\pi\)
\(29\)	−1.13917	−	1.13917i	−0.211538	−	0.211538i	0.593382	−	0.804921i	\(-0.297792\pi\)
							−0.804921	+	0.593382i	\(0.797792\pi\)
\(31\)	−3.61511	−	6.26155i	−0.649293	−	1.12461i	−0.983292	−	0.182035i	\(-0.941732\pi\)
							0.333999	−	0.942573i	\(-0.391602\pi\)
\(37\)	−2.09492	−	7.81834i	−0.344402	−	1.28533i	−0.893309	−	0.449443i	\(-0.851622\pi\)
							0.548907	−	0.835884i	\(-0.315044\pi\)
\(41\)	−2.74174			−0.428187			−0.214094	−	0.976813i	\(-0.568680\pi\)
							−0.214094	+	0.976813i	\(0.568680\pi\)
\(43\)	8.55943	+	8.55943i	1.30530	+	1.30530i	0.924767	+	0.380535i	\(0.124260\pi\)
							0.380535	+	0.924767i	\(0.375740\pi\)
\(47\)	−5.34658	+	9.26054i	−0.779878	+	1.35079i	0.152133	+	0.988360i	\(0.451386\pi\)
							−0.932011	+	0.362429i	\(0.881948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bs.a.19.16	✓	128
7.3	odd	6	inner	336.2.bs.a.115.6	yes	128
16.11	odd	4	inner	336.2.bs.a.187.6	yes	128
112.59	even	12	inner	336.2.bs.a.283.16	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bs.a.19.16	✓	128	1.1	even	1	trivial
336.2.bs.a.115.6	yes	128	7.3	odd	6	inner
336.2.bs.a.187.6	yes	128	16.11	odd	4	inner
336.2.bs.a.283.16	yes	128	112.59	even	12	inner