Embedded newform 1274.2.a.d.1.1

• Label: 1274.2.a.d.1.1
• Level: $1274$
• Weight: $2$
• Character: 1274.1
• Self dual: yes
• Analytic conductor: $10.173$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1274.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.1729412175\)
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	\(\chi\)	\(=\)	1274.1

$q$-expansion

\(f(q)\)

\(=\)

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

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	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	3.00000	1.34164	0.670820	−	0.741620i	\(-0.265942\pi\)
			0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	0	0
			\(11\)	6.00000	1.80907	0.904534	−	0.426401i	\(-0.140219\pi\)
0.904534	+	0.426401i				\(0.140219\pi\)
\(13\)	−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	\(-0.311919\pi\)
			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	\(-0.695152\pi\)
			−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	\(-0.524294\pi\)
			−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	1274.2.a.d.1.1		1
7.2	even	3		1274.2.f.r.1145.1		2
7.3	odd	6		1274.2.f.p.79.1		2
7.4	even	3		1274.2.f.r.79.1		2
7.5	odd	6		1274.2.f.p.1145.1		2
7.6	odd	2		26.2.a.a.1.1	✓	1
21.20	even	2		234.2.a.e.1.1		1
28.27	even	2		208.2.a.a.1.1		1
35.13	even	4		650.2.b.d.599.2		2
35.27	even	4		650.2.b.d.599.1		2
35.34	odd	2		650.2.a.j.1.1		1
56.13	odd	2		832.2.a.d.1.1		1
56.27	even	2		832.2.a.i.1.1		1
63.13	odd	6		2106.2.e.ba.703.1		2
63.20	even	6		2106.2.e.b.1405.1		2
63.34	odd	6		2106.2.e.ba.1405.1		2
63.41	even	6		2106.2.e.b.703.1		2
77.76	even	2		3146.2.a.n.1.1		1
84.83	odd	2		1872.2.a.q.1.1		1
91.6	even	12		338.2.e.a.23.2		4
91.20	even	12		338.2.e.a.23.1		4
91.34	even	4		338.2.b.c.337.1		2
91.41	even	12		338.2.e.a.147.1		4
91.48	odd	6		338.2.c.d.315.1		2
91.55	odd	6		338.2.c.d.191.1		2
91.62	odd	6		338.2.c.a.191.1		2
91.69	odd	6		338.2.c.a.315.1		2
91.76	even	12		338.2.e.a.147.2		4
91.83	even	4		338.2.b.c.337.2		2
91.90	odd	2		338.2.a.f.1.1		1
105.62	odd	4		5850.2.e.a.5149.2		2
105.83	odd	4		5850.2.e.a.5149.1		2
105.104	even	2		5850.2.a.p.1.1		1
112.13	odd	4		3328.2.b.m.1665.2		2
112.27	even	4		3328.2.b.j.1665.2		2
112.69	odd	4		3328.2.b.m.1665.1		2
112.83	even	4		3328.2.b.j.1665.1		2
119.118	odd	2		7514.2.a.c.1.1		1
133.132	even	2		9386.2.a.j.1.1		1
140.139	even	2		5200.2.a.x.1.1		1
168.83	odd	2		7488.2.a.h.1.1		1
168.125	even	2		7488.2.a.g.1.1		1
273.83	odd	4		3042.2.b.a.1351.1		2
273.125	odd	4		3042.2.b.a.1351.2		2
273.272	even	2		3042.2.a.a.1.1		1
364.83	odd	4		2704.2.f.d.337.2		2
364.307	odd	4		2704.2.f.d.337.1		2
364.363	even	2		2704.2.a.f.1.1		1
455.454	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	7.6	odd	2
208.2.a.a.1.1		1	28.27	even	2
234.2.a.e.1.1		1	21.20	even	2
338.2.a.f.1.1		1	91.90	odd	2
338.2.b.c.337.1		2	91.34	even	4
338.2.b.c.337.2		2	91.83	even	4
338.2.c.a.191.1		2	91.62	odd	6
338.2.c.a.315.1		2	91.69	odd	6
338.2.c.d.191.1		2	91.55	odd	6
338.2.c.d.315.1		2	91.48	odd	6
338.2.e.a.23.1		4	91.20	even	12
338.2.e.a.23.2		4	91.6	even	12
338.2.e.a.147.1		4	91.41	even	12
338.2.e.a.147.2		4	91.76	even	12
650.2.a.j.1.1		1	35.34	odd	2
650.2.b.d.599.1		2	35.27	even	4
650.2.b.d.599.2		2	35.13	even	4
832.2.a.d.1.1		1	56.13	odd	2
832.2.a.i.1.1		1	56.27	even	2
1274.2.a.d.1.1		1	1.1	even	1	trivial
1274.2.f.p.79.1		2	7.3	odd	6
1274.2.f.p.1145.1		2	7.5	odd	6
1274.2.f.r.79.1		2	7.4	even	3
1274.2.f.r.1145.1		2	7.2	even	3
1872.2.a.q.1.1		1	84.83	odd	2
2106.2.e.b.703.1		2	63.41	even	6
2106.2.e.b.1405.1		2	63.20	even	6
2106.2.e.ba.703.1		2	63.13	odd	6
2106.2.e.ba.1405.1		2	63.34	odd	6
2704.2.a.f.1.1		1	364.363	even	2
2704.2.f.d.337.1		2	364.307	odd	4
2704.2.f.d.337.2		2	364.83	odd	4
3042.2.a.a.1.1		1	273.272	even	2
3042.2.b.a.1351.1		2	273.83	odd	4
3042.2.b.a.1351.2		2	273.125	odd	4
3146.2.a.n.1.1		1	77.76	even	2
3328.2.b.j.1665.1		2	112.83	even	4
3328.2.b.j.1665.2		2	112.27	even	4
3328.2.b.m.1665.1		2	112.69	odd	4
3328.2.b.m.1665.2		2	112.13	odd	4
5200.2.a.x.1.1		1	140.139	even	2
5850.2.a.p.1.1		1	105.104	even	2
5850.2.e.a.5149.1		2	105.83	odd	4
5850.2.e.a.5149.2		2	105.62	odd	4
7488.2.a.g.1.1		1	168.125	even	2
7488.2.a.h.1.1		1	168.83	odd	2
7514.2.a.c.1.1		1	119.118	odd	2
8450.2.a.c.1.1		1	455.454	odd	2
9386.2.a.j.1.1		1	133.132	even	2