Embedded newform 3146.2.a.n.1.1

• Label: 3146.2.a.n.1.1
• Level: $3146$
• Weight: $2$
• Character: 3146.1
• Self dual: yes
• Analytic conductor: $25.121$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3146 = 2 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3146.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.1209364759\)
Analytic rank:	\(1\)
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	\(\chi\)	\(=\)	3146.1

$q$-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	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	1.00000	0.377964	0.188982	−	0.981981i	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	0	0
			\(13\)	−1.00000	−0.277350
\(17\)	3.00000	0.727607				0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	−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	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−7.00000	−1.15079	−0.575396	−	0.817875i	\(-0.695152\pi\)
			−0.575396	+	0.817875i	\(0.695152\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	1.00000	0.152499	0.0762493	−	0.997089i	\(-0.475706\pi\)
			0.0762493	+	0.997089i	\(0.475706\pi\)
\(47\)	3.00000	0.437595	0.218797	−	0.975770i	\(-0.429787\pi\)
			0.218797	+	0.975770i	\(0.429787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3146.2.a.n.1.1		1
11.10	odd	2		26.2.a.a.1.1	✓	1
33.32	even	2		234.2.a.e.1.1		1
44.43	even	2		208.2.a.a.1.1		1
55.32	even	4		650.2.b.d.599.1		2
55.43	even	4		650.2.b.d.599.2		2
55.54	odd	2		650.2.a.j.1.1		1
77.10	even	6		1274.2.f.r.79.1		2
77.32	odd	6		1274.2.f.p.79.1		2
77.54	even	6		1274.2.f.r.1145.1		2
77.65	odd	6		1274.2.f.p.1145.1		2
77.76	even	2		1274.2.a.d.1.1		1
88.21	odd	2		832.2.a.d.1.1		1
88.43	even	2		832.2.a.i.1.1		1
99.32	even	6		2106.2.e.b.703.1		2
99.43	odd	6		2106.2.e.ba.1405.1		2
99.65	even	6		2106.2.e.b.1405.1		2
99.76	odd	6		2106.2.e.ba.703.1		2
132.131	odd	2		1872.2.a.q.1.1		1
143.10	odd	6		338.2.c.a.191.1		2
143.21	even	4		338.2.b.c.337.1		2
143.32	even	12		338.2.e.a.23.2		4
143.43	odd	6		338.2.c.a.315.1		2
143.54	even	12		338.2.e.a.147.1		4
143.76	even	12		338.2.e.a.147.2		4
143.87	odd	6		338.2.c.d.315.1		2
143.98	even	12		338.2.e.a.23.1		4
143.109	even	4		338.2.b.c.337.2		2
143.120	odd	6		338.2.c.d.191.1		2
143.142	odd	2		338.2.a.f.1.1		1
165.32	odd	4		5850.2.e.a.5149.2		2
165.98	odd	4		5850.2.e.a.5149.1		2
165.164	even	2		5850.2.a.p.1.1		1
176.21	odd	4		3328.2.b.m.1665.1		2
176.43	even	4		3328.2.b.j.1665.2		2
176.109	odd	4		3328.2.b.m.1665.2		2
176.131	even	4		3328.2.b.j.1665.1		2
187.186	odd	2		7514.2.a.c.1.1		1
209.208	even	2		9386.2.a.j.1.1		1
220.219	even	2		5200.2.a.x.1.1		1
264.131	odd	2		7488.2.a.h.1.1		1
264.197	even	2		7488.2.a.g.1.1		1
429.164	odd	4		3042.2.b.a.1351.2		2
429.395	odd	4		3042.2.b.a.1351.1		2
429.428	even	2		3042.2.a.a.1.1		1
572.307	odd	4		2704.2.f.d.337.1		2
572.395	odd	4		2704.2.f.d.337.2		2
572.571	even	2		2704.2.a.f.1.1		1
715.714	odd	2		8450.2.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	11.10	odd	2
208.2.a.a.1.1		1	44.43	even	2
234.2.a.e.1.1		1	33.32	even	2
338.2.a.f.1.1		1	143.142	odd	2
338.2.b.c.337.1		2	143.21	even	4
338.2.b.c.337.2		2	143.109	even	4
338.2.c.a.191.1		2	143.10	odd	6
338.2.c.a.315.1		2	143.43	odd	6
338.2.c.d.191.1		2	143.120	odd	6
338.2.c.d.315.1		2	143.87	odd	6
338.2.e.a.23.1		4	143.98	even	12
338.2.e.a.23.2		4	143.32	even	12
338.2.e.a.147.1		4	143.54	even	12
338.2.e.a.147.2		4	143.76	even	12
650.2.a.j.1.1		1	55.54	odd	2
650.2.b.d.599.1		2	55.32	even	4
650.2.b.d.599.2		2	55.43	even	4
832.2.a.d.1.1		1	88.21	odd	2
832.2.a.i.1.1		1	88.43	even	2
1274.2.a.d.1.1		1	77.76	even	2
1274.2.f.p.79.1		2	77.32	odd	6
1274.2.f.p.1145.1		2	77.65	odd	6
1274.2.f.r.79.1		2	77.10	even	6
1274.2.f.r.1145.1		2	77.54	even	6
1872.2.a.q.1.1		1	132.131	odd	2
2106.2.e.b.703.1		2	99.32	even	6
2106.2.e.b.1405.1		2	99.65	even	6
2106.2.e.ba.703.1		2	99.76	odd	6
2106.2.e.ba.1405.1		2	99.43	odd	6
2704.2.a.f.1.1		1	572.571	even	2
2704.2.f.d.337.1		2	572.307	odd	4
2704.2.f.d.337.2		2	572.395	odd	4
3042.2.a.a.1.1		1	429.428	even	2
3042.2.b.a.1351.1		2	429.395	odd	4
3042.2.b.a.1351.2		2	429.164	odd	4
3146.2.a.n.1.1		1	1.1	even	1	trivial
3328.2.b.j.1665.1		2	176.131	even	4
3328.2.b.j.1665.2		2	176.43	even	4
3328.2.b.m.1665.1		2	176.21	odd	4
3328.2.b.m.1665.2		2	176.109	odd	4
5200.2.a.x.1.1		1	220.219	even	2
5850.2.a.p.1.1		1	165.164	even	2
5850.2.e.a.5149.1		2	165.98	odd	4
5850.2.e.a.5149.2		2	165.32	odd	4
7488.2.a.g.1.1		1	264.197	even	2
7488.2.a.h.1.1		1	264.131	odd	2
7514.2.a.c.1.1		1	187.186	odd	2
8450.2.a.c.1.1		1	715.714	odd	2
9386.2.a.j.1.1		1	209.208	even	2