Embedded newform 336.2.a.d.1.1

• Label: 336.2.a.d.1.1
• Level: $336$
• Weight: $2$
• Character: 336.1
• Self dual: yes
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	336.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)

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	0
			\(3\)	1.00000	0.577350
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	1.00000	0.377964
			\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
0.603023	+	0.797724i				\(0.293963\pi\)
\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
			0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	−8.00000	−1.66812	−0.834058	−	0.551677i	\(-0.813988\pi\)
			−0.834058	+	0.551677i	\(0.813988\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	−10.0000	−1.64399	−0.821995	−	0.569495i	\(-0.807139\pi\)
			−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.a.d.1.1		1
3.2	odd	2		1008.2.a.j.1.1		1
4.3	odd	2		42.2.a.a.1.1	✓	1
5.4	even	2		8400.2.a.k.1.1		1
7.2	even	3		2352.2.q.i.1537.1		2
7.3	odd	6		2352.2.q.n.961.1		2
7.4	even	3		2352.2.q.i.961.1		2
7.5	odd	6		2352.2.q.n.1537.1		2
7.6	odd	2		2352.2.a.l.1.1		1
8.3	odd	2		1344.2.a.q.1.1		1
8.5	even	2		1344.2.a.i.1.1		1
12.11	even	2		126.2.a.a.1.1		1
16.3	odd	4		5376.2.c.bc.2689.1		2
16.5	even	4		5376.2.c.e.2689.1		2
16.11	odd	4		5376.2.c.bc.2689.2		2
16.13	even	4		5376.2.c.e.2689.2		2
20.3	even	4		1050.2.g.a.799.1		2
20.7	even	4		1050.2.g.a.799.2		2
20.19	odd	2		1050.2.a.i.1.1		1
21.20	even	2		7056.2.a.k.1.1		1
24.5	odd	2		4032.2.a.m.1.1		1
24.11	even	2		4032.2.a.e.1.1		1
28.3	even	6		294.2.e.a.79.1		2
28.11	odd	6		294.2.e.c.79.1		2
28.19	even	6		294.2.e.a.67.1		2
28.23	odd	6		294.2.e.c.67.1		2
28.27	even	2		294.2.a.g.1.1		1
36.7	odd	6		1134.2.f.g.757.1		2
36.11	even	6		1134.2.f.j.757.1		2
36.23	even	6		1134.2.f.j.379.1		2
36.31	odd	6		1134.2.f.g.379.1		2
44.43	even	2		5082.2.a.d.1.1		1
52.51	odd	2		7098.2.a.f.1.1		1
56.13	odd	2		9408.2.a.bw.1.1		1
56.27	even	2		9408.2.a.n.1.1		1
60.23	odd	4		3150.2.g.r.2899.2		2
60.47	odd	4		3150.2.g.r.2899.1		2
60.59	even	2		3150.2.a.bo.1.1		1
84.11	even	6		882.2.g.h.667.1		2
84.23	even	6		882.2.g.h.361.1		2
84.47	odd	6		882.2.g.j.361.1		2
84.59	odd	6		882.2.g.j.667.1		2
84.83	odd	2		882.2.a.b.1.1		1
140.139	even	2		7350.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	4.3	odd	2
126.2.a.a.1.1		1	12.11	even	2
294.2.a.g.1.1		1	28.27	even	2
294.2.e.a.67.1		2	28.19	even	6
294.2.e.a.79.1		2	28.3	even	6
294.2.e.c.67.1		2	28.23	odd	6
294.2.e.c.79.1		2	28.11	odd	6
336.2.a.d.1.1		1	1.1	even	1	trivial
882.2.a.b.1.1		1	84.83	odd	2
882.2.g.h.361.1		2	84.23	even	6
882.2.g.h.667.1		2	84.11	even	6
882.2.g.j.361.1		2	84.47	odd	6
882.2.g.j.667.1		2	84.59	odd	6
1008.2.a.j.1.1		1	3.2	odd	2
1050.2.a.i.1.1		1	20.19	odd	2
1050.2.g.a.799.1		2	20.3	even	4
1050.2.g.a.799.2		2	20.7	even	4
1134.2.f.g.379.1		2	36.31	odd	6
1134.2.f.g.757.1		2	36.7	odd	6
1134.2.f.j.379.1		2	36.23	even	6
1134.2.f.j.757.1		2	36.11	even	6
1344.2.a.i.1.1		1	8.5	even	2
1344.2.a.q.1.1		1	8.3	odd	2
2352.2.a.l.1.1		1	7.6	odd	2
2352.2.q.i.961.1		2	7.4	even	3
2352.2.q.i.1537.1		2	7.2	even	3
2352.2.q.n.961.1		2	7.3	odd	6
2352.2.q.n.1537.1		2	7.5	odd	6
3150.2.a.bo.1.1		1	60.59	even	2
3150.2.g.r.2899.1		2	60.47	odd	4
3150.2.g.r.2899.2		2	60.23	odd	4
4032.2.a.e.1.1		1	24.11	even	2
4032.2.a.m.1.1		1	24.5	odd	2
5082.2.a.d.1.1		1	44.43	even	2
5376.2.c.e.2689.1		2	16.5	even	4
5376.2.c.e.2689.2		2	16.13	even	4
5376.2.c.bc.2689.1		2	16.3	odd	4
5376.2.c.bc.2689.2		2	16.11	odd	4
7056.2.a.k.1.1		1	21.20	even	2
7098.2.a.f.1.1		1	52.51	odd	2
7350.2.a.f.1.1		1	140.139	even	2
8400.2.a.k.1.1		1	5.4	even	2
9408.2.a.n.1.1		1	56.27	even	2
9408.2.a.bw.1.1		1	56.13	odd	2