Embedded newform 1764.2.a.g.1.1

• Label: 1764.2.a.g.1.1
• Level: $1764$
• Weight: $2$
• Character: 1764.1
• Self dual: yes
• Analytic conductor: $14.086$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1764.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+6.00000 q^{11} -2.00000 q^{13} +4.00000 q^{19} +6.00000 q^{23} -5.00000 q^{25} -6.00000 q^{29} -8.00000 q^{31} +2.00000 q^{37} +12.0000 q^{41} -4.00000 q^{43} +12.0000 q^{47} +6.00000 q^{53} +10.0000 q^{61} +8.00000 q^{67} -6.00000 q^{71} +10.0000 q^{73} -4.00000 q^{79} -12.0000 q^{83} +12.0000 q^{89} +10.0000 q^{97} +O(q^{100})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	0	0
			\(11\)	6.00000	1.80907	0.904534	−	0.426401i	\(-0.140219\pi\)
0.904534	+	0.426401i				\(0.140219\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	12.0000	1.87409	0.937043	−	0.349215i	\(-0.113552\pi\)
			0.937043	+	0.349215i	\(0.113552\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	12.0000	1.75038	0.875190	−	0.483779i	\(-0.160736\pi\)
			0.875190	+	0.483779i	\(0.160736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.a.g.1.1		1
3.2	odd	2		588.2.a.c.1.1		1
4.3	odd	2		7056.2.a.x.1.1		1
7.2	even	3		1764.2.k.d.361.1		2
7.3	odd	6		1764.2.k.e.1549.1		2
7.4	even	3		1764.2.k.d.1549.1		2
7.5	odd	6		1764.2.k.e.361.1		2
7.6	odd	2		252.2.a.b.1.1		1
12.11	even	2		2352.2.a.s.1.1		1
21.2	odd	6		588.2.i.f.361.1		2
21.5	even	6		588.2.i.c.361.1		2
21.11	odd	6		588.2.i.f.373.1		2
21.17	even	6		588.2.i.c.373.1		2
21.20	even	2		84.2.a.b.1.1	✓	1
24.5	odd	2		9408.2.a.co.1.1		1
24.11	even	2		9408.2.a.r.1.1		1
28.27	even	2		1008.2.a.g.1.1		1
35.13	even	4		6300.2.k.r.6049.1		2
35.27	even	4		6300.2.k.r.6049.2		2
35.34	odd	2		6300.2.a.p.1.1		1
56.13	odd	2		4032.2.a.u.1.1		1
56.27	even	2		4032.2.a.t.1.1		1
63.13	odd	6		2268.2.j.f.1513.1		2
63.20	even	6		2268.2.j.i.757.1		2
63.34	odd	6		2268.2.j.f.757.1		2
63.41	even	6		2268.2.j.i.1513.1		2
84.11	even	6		2352.2.q.g.961.1		2
84.23	even	6		2352.2.q.g.1537.1		2
84.47	odd	6		2352.2.q.s.1537.1		2
84.59	odd	6		2352.2.q.s.961.1		2
84.83	odd	2		336.2.a.b.1.1		1
105.62	odd	4		2100.2.k.a.1849.1		2
105.83	odd	4		2100.2.k.a.1849.2		2
105.104	even	2		2100.2.a.a.1.1		1
168.83	odd	2		1344.2.a.o.1.1		1
168.125	even	2		1344.2.a.f.1.1		1
336.83	odd	4		5376.2.c.x.2689.1		2
336.125	even	4		5376.2.c.i.2689.2		2
336.251	odd	4		5376.2.c.x.2689.2		2
336.293	even	4		5376.2.c.i.2689.1		2
420.419	odd	2		8400.2.a.ct.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.b.1.1	✓	1	21.20	even	2
252.2.a.b.1.1		1	7.6	odd	2
336.2.a.b.1.1		1	84.83	odd	2
588.2.a.c.1.1		1	3.2	odd	2
588.2.i.c.361.1		2	21.5	even	6
588.2.i.c.373.1		2	21.17	even	6
588.2.i.f.361.1		2	21.2	odd	6
588.2.i.f.373.1		2	21.11	odd	6
1008.2.a.g.1.1		1	28.27	even	2
1344.2.a.f.1.1		1	168.125	even	2
1344.2.a.o.1.1		1	168.83	odd	2
1764.2.a.g.1.1		1	1.1	even	1	trivial
1764.2.k.d.361.1		2	7.2	even	3
1764.2.k.d.1549.1		2	7.4	even	3
1764.2.k.e.361.1		2	7.5	odd	6
1764.2.k.e.1549.1		2	7.3	odd	6
2100.2.a.a.1.1		1	105.104	even	2
2100.2.k.a.1849.1		2	105.62	odd	4
2100.2.k.a.1849.2		2	105.83	odd	4
2268.2.j.f.757.1		2	63.34	odd	6
2268.2.j.f.1513.1		2	63.13	odd	6
2268.2.j.i.757.1		2	63.20	even	6
2268.2.j.i.1513.1		2	63.41	even	6
2352.2.a.s.1.1		1	12.11	even	2
2352.2.q.g.961.1		2	84.11	even	6
2352.2.q.g.1537.1		2	84.23	even	6
2352.2.q.s.961.1		2	84.59	odd	6
2352.2.q.s.1537.1		2	84.47	odd	6
4032.2.a.t.1.1		1	56.27	even	2
4032.2.a.u.1.1		1	56.13	odd	2
5376.2.c.i.2689.1		2	336.293	even	4
5376.2.c.i.2689.2		2	336.125	even	4
5376.2.c.x.2689.1		2	336.83	odd	4
5376.2.c.x.2689.2		2	336.251	odd	4
6300.2.a.p.1.1		1	35.34	odd	2
6300.2.k.r.6049.1		2	35.13	even	4
6300.2.k.r.6049.2		2	35.27	even	4
7056.2.a.x.1.1		1	4.3	odd	2
8400.2.a.ct.1.1		1	420.419	odd	2
9408.2.a.r.1.1		1	24.11	even	2
9408.2.a.co.1.1		1	24.5	odd	2