Embedded newform 336.2.bs.a.19.12

• Label: 336.2.bs.a.19.12
• 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			19.12
Character	\(\chi\)	\(=\)	336.19
Dual form			336.2.bs.a.283.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.867553 + 1.11685i) q^{2} +(-0.258819 + 0.965926i) q^{3} +(-0.494702 - 1.93785i) q^{4} +(1.83632 - 0.492041i) q^{5} +(-0.854254 - 1.12705i) q^{6} +(-1.58583 - 2.11781i) q^{7} +(2.59347 + 1.12868i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 4 q^{4} - 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.867553	+	1.11685i	−0.613453	+	0.789731i
							\(3\)	−0.258819	+	0.965926i	−0.149429	+	0.557678i
\(5\)	1.83632	−	0.492041i	0.821228	−	0.220047i								0.176345	−	0.984328i	\(-0.443573\pi\)
							0.644883	+	0.764281i	\(0.276906\pi\)
\(7\)	−1.58583	−	2.11781i	−0.599389	−	0.800458i
							\(11\)	0.978501	−	3.65181i	0.295029	−	1.10106i	−0.646165	−	0.763198i	\(-0.723628\pi\)
0.941194	−	0.337866i	\(-0.109705\pi\)
\(13\)	3.02000	−	3.02000i	0.837599	−	0.837599i	−0.150944	−	0.988542i	\(-0.548231\pi\)
							0.988542	+	0.150944i	\(0.0482312\pi\)
\(17\)	2.32244	−	1.34086i	0.563275	−	0.325207i	−0.191184	−	0.981554i	\(-0.561233\pi\)
							0.754459	+	0.656347i	\(0.227899\pi\)
\(19\)	−2.64736	+	0.709357i	−0.607345	+	0.162738i	−0.549368	−	0.835580i	\(-0.685132\pi\)
							−0.0579767	+	0.998318i	\(0.518465\pi\)
\(23\)	2.00507	−	3.47288i	0.418085	−	0.724145i	−0.577661	−	0.816276i	\(-0.696035\pi\)
							0.995747	+	0.0921312i	\(0.0293679\pi\)
\(29\)	3.59395	+	3.59395i	0.667379	+	0.667379i	0.957109	−	0.289729i	\(-0.0935653\pi\)
							−0.289729	+	0.957109i	\(0.593565\pi\)
\(31\)	2.17317	+	3.76403i	0.390312	+	0.676041i	0.992491	−	0.122321i	\(-0.0390337\pi\)
							−0.602178	+	0.798362i	\(0.705700\pi\)
\(37\)	1.10383	+	4.11953i	0.181468	+	0.677247i	0.995359	+	0.0962299i	\(0.0306784\pi\)
							−0.813891	+	0.581017i	\(0.802655\pi\)
\(41\)	8.30654			1.29726			0.648632	−	0.761103i	\(-0.275342\pi\)
							0.648632	+	0.761103i	\(0.275342\pi\)
\(43\)	−1.87127	−	1.87127i	−0.285367	−	0.285367i	0.549878	−	0.835245i	\(-0.314674\pi\)
							−0.835245	+	0.549878i	\(0.814674\pi\)
\(47\)	−0.663881	+	1.14988i	−0.0968370	+	0.167727i	−0.910374	−	0.413787i	\(-0.864206\pi\)
							0.813537	+	0.581513i	\(0.197539\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bs.a.19.12	✓	128
7.3	odd	6	inner	336.2.bs.a.115.12	yes	128
16.11	odd	4	inner	336.2.bs.a.187.12	yes	128
112.59	even	12	inner	336.2.bs.a.283.12	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bs.a.19.12	✓	128	1.1	even	1	trivial
336.2.bs.a.115.12	yes	128	7.3	odd	6	inner
336.2.bs.a.187.12	yes	128	16.11	odd	4	inner
336.2.bs.a.283.12	yes	128	112.59	even	12	inner