Embedded newform 338.2.a.e.1.1

• Label: 338.2.a.e.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

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +4.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -3.00000 q^{18} -1.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} +4.00000 q^{28} -1.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -4.00000 q^{35} -3.00000 q^{36} +3.00000 q^{37} -1.00000 q^{40} -9.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -4.00000 q^{50} -9.00000 q^{53} -4.00000 q^{55} +4.00000 q^{56} -1.00000 q^{58} -4.00000 q^{59} +7.00000 q^{61} +4.00000 q^{62} -12.0000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +3.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} +11.0000 q^{73} +3.00000 q^{74} +16.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -9.00000 q^{82} -3.00000 q^{85} -8.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -4.00000 q^{92} -8.00000 q^{94} +2.00000 q^{97} +9.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
			\(3\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	−1.00000	−0.447214	−0.223607	−	0.974679i	\(-0.571783\pi\)
			−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	0	0
			\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
0.363803	+	0.931476i				\(0.381478\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	−1.00000	−0.185695	−0.0928477	−	0.995680i	\(-0.529597\pi\)
			−0.0928477	+	0.995680i	\(0.529597\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	3.00000	0.493197	0.246598	−	0.969118i	\(-0.420687\pi\)
			0.246598	+	0.969118i	\(0.420687\pi\)
\(41\)	−9.00000	−1.40556	−0.702782	−	0.711405i	\(-0.748059\pi\)
			−0.702782	+	0.711405i	\(0.748059\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.2.a.e.1.1		1
3.2	odd	2		3042.2.a.e.1.1		1
4.3	odd	2		2704.2.a.h.1.1		1
5.4	even	2		8450.2.a.f.1.1		1
13.2	odd	12		338.2.e.b.147.2		4
13.3	even	3		26.2.c.a.9.1	yes	2
13.4	even	6		338.2.c.e.315.1		2
13.5	odd	4		338.2.b.b.337.1		2
13.6	odd	12		338.2.e.b.23.1		4
13.7	odd	12		338.2.e.b.23.2		4
13.8	odd	4		338.2.b.b.337.2		2
13.9	even	3		26.2.c.a.3.1	✓	2
13.10	even	6		338.2.c.e.191.1		2
13.11	odd	12		338.2.e.b.147.1		4
13.12	even	2		338.2.a.c.1.1		1
39.5	even	4		3042.2.b.e.1351.2		2
39.8	even	4		3042.2.b.e.1351.1		2
39.29	odd	6		234.2.h.c.217.1		2
39.35	odd	6		234.2.h.c.55.1		2
39.38	odd	2		3042.2.a.k.1.1		1
52.3	odd	6		208.2.i.b.113.1		2
52.31	even	4		2704.2.f.g.337.2		2
52.35	odd	6		208.2.i.b.81.1		2
52.47	even	4		2704.2.f.g.337.1		2
52.51	odd	2		2704.2.a.i.1.1		1
65.3	odd	12		650.2.o.c.399.2		4
65.9	even	6		650.2.e.c.601.1		2
65.22	odd	12		650.2.o.c.549.2		4
65.29	even	6		650.2.e.c.451.1		2
65.42	odd	12		650.2.o.c.399.1		4
65.48	odd	12		650.2.o.c.549.1		4
65.64	even	2		8450.2.a.s.1.1		1
91.3	odd	6		1274.2.h.a.373.1		2
91.9	even	3		1274.2.h.b.263.1		2
91.16	even	3		1274.2.e.n.165.1		2
91.48	odd	6		1274.2.g.a.393.1		2
91.55	odd	6		1274.2.g.a.295.1		2
91.61	odd	6		1274.2.h.a.263.1		2
91.68	odd	6		1274.2.e.m.165.1		2
91.74	even	3		1274.2.e.n.471.1		2
91.81	even	3		1274.2.h.b.373.1		2
91.87	odd	6		1274.2.e.m.471.1		2
104.3	odd	6		832.2.i.f.321.1		2
104.29	even	6		832.2.i.e.321.1		2
104.35	odd	6		832.2.i.f.705.1		2
104.61	even	6		832.2.i.e.705.1		2
156.35	even	6		1872.2.t.k.289.1		2
156.107	even	6		1872.2.t.k.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.c.a.3.1	✓	2	13.9	even	3
26.2.c.a.9.1	yes	2	13.3	even	3
208.2.i.b.81.1		2	52.35	odd	6
208.2.i.b.113.1		2	52.3	odd	6
234.2.h.c.55.1		2	39.35	odd	6
234.2.h.c.217.1		2	39.29	odd	6
338.2.a.c.1.1		1	13.12	even	2
338.2.a.e.1.1		1	1.1	even	1	trivial
338.2.b.b.337.1		2	13.5	odd	4
338.2.b.b.337.2		2	13.8	odd	4
338.2.c.e.191.1		2	13.10	even	6
338.2.c.e.315.1		2	13.4	even	6
338.2.e.b.23.1		4	13.6	odd	12
338.2.e.b.23.2		4	13.7	odd	12
338.2.e.b.147.1		4	13.11	odd	12
338.2.e.b.147.2		4	13.2	odd	12
650.2.e.c.451.1		2	65.29	even	6
650.2.e.c.601.1		2	65.9	even	6
650.2.o.c.399.1		4	65.42	odd	12
650.2.o.c.399.2		4	65.3	odd	12
650.2.o.c.549.1		4	65.48	odd	12
650.2.o.c.549.2		4	65.22	odd	12
832.2.i.e.321.1		2	104.29	even	6
832.2.i.e.705.1		2	104.61	even	6
832.2.i.f.321.1		2	104.3	odd	6
832.2.i.f.705.1		2	104.35	odd	6
1274.2.e.m.165.1		2	91.68	odd	6
1274.2.e.m.471.1		2	91.87	odd	6
1274.2.e.n.165.1		2	91.16	even	3
1274.2.e.n.471.1		2	91.74	even	3
1274.2.g.a.295.1		2	91.55	odd	6
1274.2.g.a.393.1		2	91.48	odd	6
1274.2.h.a.263.1		2	91.61	odd	6
1274.2.h.a.373.1		2	91.3	odd	6
1274.2.h.b.263.1		2	91.9	even	3
1274.2.h.b.373.1		2	91.81	even	3
1872.2.t.k.289.1		2	156.35	even	6
1872.2.t.k.1153.1		2	156.107	even	6
2704.2.a.h.1.1		1	4.3	odd	2
2704.2.a.i.1.1		1	52.51	odd	2
2704.2.f.g.337.1		2	52.47	even	4
2704.2.f.g.337.2		2	52.31	even	4
3042.2.a.e.1.1		1	3.2	odd	2
3042.2.a.k.1.1		1	39.38	odd	2
3042.2.b.e.1351.1		2	39.8	even	4
3042.2.b.e.1351.2		2	39.5	even	4
8450.2.a.f.1.1		1	5.4	even	2
8450.2.a.s.1.1		1	65.64	even	2