Embedded newform 338.2.a.f.1.1

• Label: 338.2.a.f.1.1
• Level: $338$
• Weight: $2$
• Character: 338.1
• Self dual: yes
• Analytic conductor: $2.699$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.69894358832\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	338.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	3.00000	1.34164	0.670820	−	0.741620i	\(-0.265942\pi\)
			0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	1.00000	0.377964	0.188982	−	0.981981i	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	−6.00000	−1.80907	−0.904534	−	0.426401i	\(-0.859781\pi\)
			−0.904534	+	0.426401i	\(0.859781\pi\)
\(13\)	0	0
			\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
−0.363803	+	0.931476i				\(0.618522\pi\)
\(19\)	−2.00000	−0.458831	−0.229416	−	0.973329i	\(-0.573682\pi\)
			−0.229416	+	0.973329i	\(0.573682\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	7.00000	1.15079	0.575396	−	0.817875i	\(-0.304848\pi\)
			0.575396	+	0.817875i	\(0.304848\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	−1.00000	−0.152499	−0.0762493	−	0.997089i	\(-0.524294\pi\)
			−0.0762493	+	0.997089i	\(0.524294\pi\)
\(47\)	−3.00000	−0.437595	−0.218797	−	0.975770i	\(-0.570213\pi\)
			−0.218797	+	0.975770i	\(0.570213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.2.a.f.1.1		1
3.2	odd	2		3042.2.a.a.1.1		1
4.3	odd	2		2704.2.a.f.1.1		1
5.4	even	2		8450.2.a.c.1.1		1
13.2	odd	12		338.2.e.a.147.2		4
13.3	even	3		338.2.c.a.191.1		2
13.4	even	6		338.2.c.d.315.1		2
13.5	odd	4		338.2.b.c.337.1		2
13.6	odd	12		338.2.e.a.23.1		4
13.7	odd	12		338.2.e.a.23.2		4
13.8	odd	4		338.2.b.c.337.2		2
13.9	even	3		338.2.c.a.315.1		2
13.10	even	6		338.2.c.d.191.1		2
13.11	odd	12		338.2.e.a.147.1		4
13.12	even	2		26.2.a.a.1.1	✓	1
39.5	even	4		3042.2.b.a.1351.2		2
39.8	even	4		3042.2.b.a.1351.1		2
39.38	odd	2		234.2.a.e.1.1		1
52.31	even	4		2704.2.f.d.337.1		2
52.47	even	4		2704.2.f.d.337.2		2
52.51	odd	2		208.2.a.a.1.1		1
65.12	odd	4		650.2.b.d.599.1		2
65.38	odd	4		650.2.b.d.599.2		2
65.64	even	2		650.2.a.j.1.1		1
91.12	odd	6		1274.2.f.r.1145.1		2
91.25	even	6		1274.2.f.p.79.1		2
91.38	odd	6		1274.2.f.r.79.1		2
91.51	even	6		1274.2.f.p.1145.1		2
91.90	odd	2		1274.2.a.d.1.1		1
104.51	odd	2		832.2.a.i.1.1		1
104.77	even	2		832.2.a.d.1.1		1
117.25	even	6		2106.2.e.ba.1405.1		2
117.38	odd	6		2106.2.e.b.1405.1		2
117.77	odd	6		2106.2.e.b.703.1		2
117.103	even	6		2106.2.e.ba.703.1		2
143.142	odd	2		3146.2.a.n.1.1		1
156.155	even	2		1872.2.a.q.1.1		1
195.38	even	4		5850.2.e.a.5149.1		2
195.77	even	4		5850.2.e.a.5149.2		2
195.194	odd	2		5850.2.a.p.1.1		1
208.51	odd	4		3328.2.b.j.1665.1		2
208.77	even	4		3328.2.b.m.1665.2		2
208.155	odd	4		3328.2.b.j.1665.2		2
208.181	even	4		3328.2.b.m.1665.1		2
221.220	even	2		7514.2.a.c.1.1		1
247.246	odd	2		9386.2.a.j.1.1		1
260.259	odd	2		5200.2.a.x.1.1		1
312.77	odd	2		7488.2.a.g.1.1		1
312.155	even	2		7488.2.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	13.12	even	2
208.2.a.a.1.1		1	52.51	odd	2
234.2.a.e.1.1		1	39.38	odd	2
338.2.a.f.1.1		1	1.1	even	1	trivial
338.2.b.c.337.1		2	13.5	odd	4
338.2.b.c.337.2		2	13.8	odd	4
338.2.c.a.191.1		2	13.3	even	3
338.2.c.a.315.1		2	13.9	even	3
338.2.c.d.191.1		2	13.10	even	6
338.2.c.d.315.1		2	13.4	even	6
338.2.e.a.23.1		4	13.6	odd	12
338.2.e.a.23.2		4	13.7	odd	12
338.2.e.a.147.1		4	13.11	odd	12
338.2.e.a.147.2		4	13.2	odd	12
650.2.a.j.1.1		1	65.64	even	2
650.2.b.d.599.1		2	65.12	odd	4
650.2.b.d.599.2		2	65.38	odd	4
832.2.a.d.1.1		1	104.77	even	2
832.2.a.i.1.1		1	104.51	odd	2
1274.2.a.d.1.1		1	91.90	odd	2
1274.2.f.p.79.1		2	91.25	even	6
1274.2.f.p.1145.1		2	91.51	even	6
1274.2.f.r.79.1		2	91.38	odd	6
1274.2.f.r.1145.1		2	91.12	odd	6
1872.2.a.q.1.1		1	156.155	even	2
2106.2.e.b.703.1		2	117.77	odd	6
2106.2.e.b.1405.1		2	117.38	odd	6
2106.2.e.ba.703.1		2	117.103	even	6
2106.2.e.ba.1405.1		2	117.25	even	6
2704.2.a.f.1.1		1	4.3	odd	2
2704.2.f.d.337.1		2	52.31	even	4
2704.2.f.d.337.2		2	52.47	even	4
3042.2.a.a.1.1		1	3.2	odd	2
3042.2.b.a.1351.1		2	39.8	even	4
3042.2.b.a.1351.2		2	39.5	even	4
3146.2.a.n.1.1		1	143.142	odd	2
3328.2.b.j.1665.1		2	208.51	odd	4
3328.2.b.j.1665.2		2	208.155	odd	4
3328.2.b.m.1665.1		2	208.181	even	4
3328.2.b.m.1665.2		2	208.77	even	4
5200.2.a.x.1.1		1	260.259	odd	2
5850.2.a.p.1.1		1	195.194	odd	2
5850.2.e.a.5149.1		2	195.38	even	4
5850.2.e.a.5149.2		2	195.77	even	4
7488.2.a.g.1.1		1	312.77	odd	2
7488.2.a.h.1.1		1	312.155	even	2
7514.2.a.c.1.1		1	221.220	even	2
8450.2.a.c.1.1		1	5.4	even	2
9386.2.a.j.1.1		1	247.246	odd	2