Embedded newform 1764.2.a.a.1.1

• Label: 1764.2.a.a.1.1
• Level: $1764$
• Weight: $2$
• Character: 1764.1
• Self dual: yes
• Analytic conductor: $14.086$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{5} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	−3.00000	−1.34164				−0.670820	−	0.741620i	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	0	0
			\(11\)	3.00000	0.904534	0.452267	−	0.891883i	\(-0.350615\pi\)
0.452267	+	0.891883i				\(0.350615\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
			−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	−1.00000	−0.229416	−0.114708	−	0.993399i	\(-0.536593\pi\)
			−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	−3.00000	−0.625543	−0.312772	−	0.949828i	\(-0.601257\pi\)
			−0.312772	+	0.949828i	\(0.601257\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−7.00000	−1.25724	−0.628619	−	0.777714i	\(-0.716379\pi\)
			−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	−1.00000	−0.164399	−0.0821995	−	0.996616i	\(-0.526194\pi\)
			−0.0821995	+	0.996616i	\(0.526194\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	9.00000	1.31278	0.656392	−	0.754420i	\(-0.272082\pi\)
			0.656392	+	0.754420i	\(0.272082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.a.a.1.1		1
3.2	odd	2		196.2.a.b.1.1		1
4.3	odd	2		7056.2.a.f.1.1		1
7.2	even	3		252.2.k.c.109.1		2
7.3	odd	6		1764.2.k.b.1549.1		2
7.4	even	3		252.2.k.c.37.1		2
7.5	odd	6		1764.2.k.b.361.1		2
7.6	odd	2		1764.2.a.j.1.1		1
12.11	even	2		784.2.a.d.1.1		1
15.2	even	4		4900.2.e.i.2549.1		2
15.8	even	4		4900.2.e.i.2549.2		2
15.14	odd	2		4900.2.a.g.1.1		1
21.2	odd	6		28.2.e.a.25.1	yes	2
21.5	even	6		196.2.e.a.165.1		2
21.11	odd	6		28.2.e.a.9.1	✓	2
21.17	even	6		196.2.e.a.177.1		2
21.20	even	2		196.2.a.a.1.1		1
24.5	odd	2		3136.2.a.h.1.1		1
24.11	even	2		3136.2.a.s.1.1		1
28.11	odd	6		1008.2.s.p.289.1		2
28.23	odd	6		1008.2.s.p.865.1		2
28.27	even	2		7056.2.a.bw.1.1		1
63.2	odd	6		2268.2.l.h.109.1		2
63.4	even	3		2268.2.l.a.541.1		2
63.11	odd	6		2268.2.i.a.2053.1		2
63.16	even	3		2268.2.l.a.109.1		2
63.23	odd	6		2268.2.i.a.865.1		2
63.25	even	3		2268.2.i.h.2053.1		2
63.32	odd	6		2268.2.l.h.541.1		2
63.58	even	3		2268.2.i.h.865.1		2
84.11	even	6		112.2.i.b.65.1		2
84.23	even	6		112.2.i.b.81.1		2
84.47	odd	6		784.2.i.d.753.1		2
84.59	odd	6		784.2.i.d.177.1		2
84.83	odd	2		784.2.a.g.1.1		1
105.2	even	12		700.2.r.b.249.2		4
105.23	even	12		700.2.r.b.249.1		4
105.32	even	12		700.2.r.b.149.1		4
105.44	odd	6		700.2.i.c.501.1		2
105.53	even	12		700.2.r.b.149.2		4
105.62	odd	4		4900.2.e.h.2549.2		2
105.74	odd	6		700.2.i.c.401.1		2
105.83	odd	4		4900.2.e.h.2549.1		2
105.104	even	2		4900.2.a.n.1.1		1
168.11	even	6		448.2.i.c.65.1		2
168.53	odd	6		448.2.i.e.65.1		2
168.83	odd	2		3136.2.a.k.1.1		1
168.107	even	6		448.2.i.c.193.1		2
168.125	even	2		3136.2.a.v.1.1		1
168.149	odd	6		448.2.i.e.193.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	21.11	odd	6
28.2.e.a.25.1	yes	2	21.2	odd	6
112.2.i.b.65.1		2	84.11	even	6
112.2.i.b.81.1		2	84.23	even	6
196.2.a.a.1.1		1	21.20	even	2
196.2.a.b.1.1		1	3.2	odd	2
196.2.e.a.165.1		2	21.5	even	6
196.2.e.a.177.1		2	21.17	even	6
252.2.k.c.37.1		2	7.4	even	3
252.2.k.c.109.1		2	7.2	even	3
448.2.i.c.65.1		2	168.11	even	6
448.2.i.c.193.1		2	168.107	even	6
448.2.i.e.65.1		2	168.53	odd	6
448.2.i.e.193.1		2	168.149	odd	6
700.2.i.c.401.1		2	105.74	odd	6
700.2.i.c.501.1		2	105.44	odd	6
700.2.r.b.149.1		4	105.32	even	12
700.2.r.b.149.2		4	105.53	even	12
700.2.r.b.249.1		4	105.23	even	12
700.2.r.b.249.2		4	105.2	even	12
784.2.a.d.1.1		1	12.11	even	2
784.2.a.g.1.1		1	84.83	odd	2
784.2.i.d.177.1		2	84.59	odd	6
784.2.i.d.753.1		2	84.47	odd	6
1008.2.s.p.289.1		2	28.11	odd	6
1008.2.s.p.865.1		2	28.23	odd	6
1764.2.a.a.1.1		1	1.1	even	1	trivial
1764.2.a.j.1.1		1	7.6	odd	2
1764.2.k.b.361.1		2	7.5	odd	6
1764.2.k.b.1549.1		2	7.3	odd	6
2268.2.i.a.865.1		2	63.23	odd	6
2268.2.i.a.2053.1		2	63.11	odd	6
2268.2.i.h.865.1		2	63.58	even	3
2268.2.i.h.2053.1		2	63.25	even	3
2268.2.l.a.109.1		2	63.16	even	3
2268.2.l.a.541.1		2	63.4	even	3
2268.2.l.h.109.1		2	63.2	odd	6
2268.2.l.h.541.1		2	63.32	odd	6
3136.2.a.h.1.1		1	24.5	odd	2
3136.2.a.k.1.1		1	168.83	odd	2
3136.2.a.s.1.1		1	24.11	even	2
3136.2.a.v.1.1		1	168.125	even	2
4900.2.a.g.1.1		1	15.14	odd	2
4900.2.a.n.1.1		1	105.104	even	2
4900.2.e.h.2549.1		2	105.83	odd	4
4900.2.e.h.2549.2		2	105.62	odd	4
4900.2.e.i.2549.1		2	15.2	even	4
4900.2.e.i.2549.2		2	15.8	even	4
7056.2.a.f.1.1		1	4.3	odd	2
7056.2.a.bw.1.1		1	28.27	even	2