Embedded newform 336.2.bu.a.275.29

• Label: 336.2.bu.a.275.29
• Level: $336$
• Weight: $2$
• Character: 336.275
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			275.29
Character	\(\chi\)	\(=\)	336.275
Dual form			336.2.bu.a.11.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0637563 - 1.41278i) q^{2} +(1.41400 + 1.00031i) q^{3} +(-1.99187 + 0.180147i) q^{4} +(-2.20205 + 0.590036i) q^{5} +(1.32306 - 2.06143i) q^{6} +(-1.05645 + 2.42568i) q^{7} +(0.381501 + 2.80258i) q^{8} +(0.998765 + 2.82886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 2 q^{3} - 4 q^{4} - 8 q^{6} - 16 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0637563	−	1.41278i	−0.0450825	−	0.998983i
							\(3\)	1.41400	+	1.00031i	0.816370	+	0.577529i
\(5\)	−2.20205	+	0.590036i	−0.984785	+	0.263872i								−0.715058	−	0.699065i	\(-0.753600\pi\)
							−0.269726	+	0.962937i	\(0.586933\pi\)
\(7\)	−1.05645	+	2.42568i	−0.399300	+	0.916820i
							\(11\)	1.64179	+	0.439916i	0.495018	+	0.132640i	0.497687	−	0.867357i	\(-0.334183\pi\)
−0.00266871	+	0.999996i	\(0.500849\pi\)
\(13\)	3.41198	+	3.41198i	0.946313	+	0.946313i	0.998630	−	0.0523177i	\(-0.0166608\pi\)
							−0.0523177	+	0.998630i	\(0.516661\pi\)
\(17\)	0.912798	−	0.527004i	0.221386	−	0.127817i	−0.385206	−	0.922831i	\(-0.625870\pi\)
							0.606592	+	0.795013i	\(0.292536\pi\)
\(19\)	−0.612674	−	2.28653i	−0.140557	−	0.524566i	−0.999913	−	0.0131881i	\(-0.995802\pi\)
							0.859356	−	0.511378i	\(-0.170865\pi\)
\(23\)	0.394710	+	0.227886i	0.0823027	+	0.0475175i	0.540587	−	0.841288i	\(-0.318202\pi\)
							−0.458284	+	0.888806i	\(0.651536\pi\)
\(29\)	−4.79632	+	4.79632i	−0.890655	+	0.890655i	−0.994585	−	0.103930i	\(-0.966858\pi\)
							0.103930	+	0.994585i	\(0.466858\pi\)
\(31\)	−4.40906	+	2.54557i	−0.791890	+	0.457198i	−0.840627	−	0.541614i	\(-0.817814\pi\)
							0.0487376	+	0.998812i	\(0.484480\pi\)
\(37\)	−1.89714	−	7.08023i	−0.311888	−	1.16398i	−0.926852	−	0.375427i	\(-0.877496\pi\)
							0.614964	−	0.788556i	\(-0.289171\pi\)
\(41\)	11.1908			1.74770			0.873851	−	0.486194i	\(-0.161615\pi\)
							0.873851	+	0.486194i	\(0.161615\pi\)
\(43\)	−3.32912	−	3.32912i	−0.507686	−	0.507686i	0.406129	−	0.913816i	\(-0.366878\pi\)
							−0.913816	+	0.406129i	\(0.866878\pi\)
\(47\)	−3.06628	+	5.31095i	−0.447263	+	0.774682i	−0.998207	−	0.0598604i	\(-0.980934\pi\)
							0.550944	+	0.834542i	\(0.314268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bu.a.275.29	yes	240
3.2	odd	2	inner	336.2.bu.a.275.32	yes	240
7.4	even	3	inner	336.2.bu.a.179.51	yes	240
16.11	odd	4	inner	336.2.bu.a.107.10	yes	240
21.11	odd	6	inner	336.2.bu.a.179.10	yes	240
48.11	even	4	inner	336.2.bu.a.107.51	yes	240
112.11	odd	12	inner	336.2.bu.a.11.32	yes	240
336.11	even	12	inner	336.2.bu.a.11.29	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bu.a.11.29	✓	240	336.11	even	12	inner
336.2.bu.a.11.32	yes	240	112.11	odd	12	inner
336.2.bu.a.107.10	yes	240	16.11	odd	4	inner
336.2.bu.a.107.51	yes	240	48.11	even	4	inner
336.2.bu.a.179.10	yes	240	21.11	odd	6	inner
336.2.bu.a.179.51	yes	240	7.4	even	3	inner
336.2.bu.a.275.29	yes	240	1.1	even	1	trivial
336.2.bu.a.275.32	yes	240	3.2	odd	2	inner