Embedded newform 336.2.bq.b.37.1

• Label: 336.2.bq.b.37.1
• Level: $336$
• Weight: $2$
• Character: 336.37
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.1
Character	\(\chi\)	\(=\)	336.37
Dual form			336.2.bq.b.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39055 - 0.257600i) q^{2} +(0.965926 + 0.258819i) q^{3} +(1.86728 + 0.716415i) q^{4} +(-0.588887 + 0.157792i) q^{5} +(-1.27650 - 0.608725i) q^{6} +(2.58295 - 0.573024i) q^{7} +(-2.41201 - 1.47723i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39055	−	0.257600i	−0.983271	−	0.182151i
							\(3\)	0.965926	+	0.258819i	0.557678	+	0.149429i
\(5\)	−0.588887	+	0.157792i	−0.263358	+	0.0705666i								−0.388082	−	0.921625i	\(-0.626862\pi\)
							0.124724	+	0.992192i	\(0.460196\pi\)
\(7\)	2.58295	−	0.573024i	0.976264	−	0.216583i
							\(11\)	0.0594702	−	0.221946i	0.0179309	−	0.0669191i	−0.956381	−	0.292123i	\(-0.905638\pi\)
0.974312	+	0.225204i	\(0.0723049\pi\)
\(13\)	1.86885	−	1.86885i	0.518325	−	0.518325i	−0.398739	−	0.917064i	\(-0.630552\pi\)
							0.917064	+	0.398739i	\(0.130552\pi\)
\(17\)	1.70915	+	2.96034i	0.414530	+	0.717987i	0.995379	−	0.0960241i	\(-0.0306126\pi\)
							−0.580849	+	0.814011i	\(0.697279\pi\)
\(19\)	−0.417729	−	1.55898i	−0.0958335	−	0.357656i	0.901311	−	0.433173i	\(-0.142606\pi\)
							−0.997145	+	0.0755170i	\(0.975939\pi\)
\(23\)	2.65039	+	1.53020i	0.552645	+	0.319069i	0.750188	−	0.661225i	\(-0.229963\pi\)
							−0.197543	+	0.980294i	\(0.563296\pi\)
\(29\)	4.50223	−	4.50223i	0.836043	−	0.836043i	−0.152292	−	0.988335i	\(-0.548666\pi\)
							0.988335	+	0.152292i	\(0.0486655\pi\)
\(31\)	2.55841	+	4.43129i	0.459504	+	0.795884i	0.998935	−	0.0461461i	\(-0.0146940\pi\)
							−0.539431	+	0.842030i	\(0.681361\pi\)
\(37\)	5.51046	−	1.47652i	0.905913	−	0.242739i	0.224359	−	0.974507i	\(-0.427971\pi\)
							0.681554	+	0.731768i	\(0.261304\pi\)
\(41\)		−	6.64364i		−	1.03756i	−0.854907	−	0.518781i	\(-0.826386\pi\)
							0.854907	−	0.518781i	\(-0.173614\pi\)
\(43\)	1.27101	+	1.27101i	0.193827	+	0.193827i	0.797348	−	0.603520i	\(-0.206236\pi\)
							−0.603520	+	0.797348i	\(0.706236\pi\)
\(47\)	−3.46555	+	6.00251i	−0.505503	+	0.875557i	0.494477	+	0.869191i	\(0.335360\pi\)
							−0.999980	+	0.00636583i	\(0.997974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.37.1	✓	120
7.4	even	3	inner	336.2.bq.b.277.22	yes	120
16.13	even	4	inner	336.2.bq.b.205.22	yes	120
112.109	even	12	inner	336.2.bq.b.109.1	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.1	✓	120	1.1	even	1	trivial
336.2.bq.b.109.1	yes	120	112.109	even	12	inner
336.2.bq.b.205.22	yes	120	16.13	even	4	inner
336.2.bq.b.277.22	yes	120	7.4	even	3	inner