Embedded newform 1728.4.a.k.1.1

• Label: 1728.4.a.k.1.1
• Level: $1728$
• Weight: $4$
• Character: 1728.1
• Self dual: yes
• Analytic conductor: $101.955$
• Analytic rank: $2$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(101.955300490\)
Analytic rank:	\(2\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1728.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{5} -29.0000 q^{7} +O(q^{10})\)

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	−0.268328				−0.134164	−	0.990959i	\(-0.542835\pi\)
			−0.134164	+	0.990959i	\(0.542835\pi\)
\(7\)	−29.0000	−1.56585	−0.782926	−	0.622114i	\(-0.786274\pi\)
			−0.782926	+	0.622114i	\(0.786274\pi\)
\(11\)	−57.0000	−1.56238	−0.781188	−	0.624295i	\(-0.785386\pi\)
			−0.781188	+	0.624295i	\(0.785386\pi\)
\(13\)	−20.0000	−0.426692	−0.213346	−	0.976977i	\(-0.568436\pi\)
			−0.213346	+	0.976977i	\(0.568436\pi\)
\(17\)	−72.0000	−1.02721	−0.513605	−	0.858027i	\(-0.671690\pi\)
			−0.513605	+	0.858027i	\(0.671690\pi\)
\(19\)	−106.000	−1.27990	−0.639949	−	0.768417i	\(-0.721045\pi\)
			−0.639949	+	0.768417i	\(0.721045\pi\)
\(23\)	−174.000	−1.57746	−0.788728	−	0.614742i	\(-0.789260\pi\)
			−0.788728	+	0.614742i	\(0.789260\pi\)
\(29\)	210.000	1.34469	0.672345	−	0.740238i	\(-0.265287\pi\)
			0.672345	+	0.740238i	\(0.265287\pi\)
\(31\)	−47.0000	−0.272305	−0.136152	−	0.990688i	\(-0.543474\pi\)
			−0.136152	+	0.990688i	\(0.543474\pi\)
\(37\)	−2.00000	−0.00888643	−0.00444322	−	0.999990i	\(-0.501414\pi\)
			−0.00444322	+	0.999990i	\(0.501414\pi\)
\(41\)	−6.00000	−0.0228547	−0.0114273	−	0.999935i	\(-0.503638\pi\)
			−0.0114273	+	0.999935i	\(0.503638\pi\)
\(43\)	218.000	0.773132	0.386566	−	0.922262i	\(-0.373661\pi\)
			0.386566	+	0.922262i	\(0.373661\pi\)
\(47\)	−474.000	−1.47106	−0.735532	−	0.677490i	\(-0.763068\pi\)
			−0.735532	+	0.677490i	\(0.763068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.k.1.1		1
3.2	odd	2		1728.4.a.u.1.1		1
4.3	odd	2		1728.4.a.l.1.1		1
8.3	odd	2		54.4.a.c.1.1	yes	1
8.5	even	2		432.4.a.j.1.1		1
12.11	even	2		1728.4.a.v.1.1		1
24.5	odd	2		432.4.a.e.1.1		1
24.11	even	2		54.4.a.b.1.1	✓	1
40.3	even	4		1350.4.c.b.649.1		2
40.19	odd	2		1350.4.a.a.1.1		1
40.27	even	4		1350.4.c.b.649.2		2
72.11	even	6		162.4.c.g.109.1		2
72.43	odd	6		162.4.c.b.109.1		2
72.59	even	6		162.4.c.g.55.1		2
72.67	odd	6		162.4.c.b.55.1		2
120.59	even	2		1350.4.a.o.1.1		1
120.83	odd	4		1350.4.c.s.649.2		2
120.107	odd	4		1350.4.c.s.649.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.a.b.1.1	✓	1	24.11	even	2
54.4.a.c.1.1	yes	1	8.3	odd	2
162.4.c.b.55.1		2	72.67	odd	6
162.4.c.b.109.1		2	72.43	odd	6
162.4.c.g.55.1		2	72.59	even	6
162.4.c.g.109.1		2	72.11	even	6
432.4.a.e.1.1		1	24.5	odd	2
432.4.a.j.1.1		1	8.5	even	2
1350.4.a.a.1.1		1	40.19	odd	2
1350.4.a.o.1.1		1	120.59	even	2
1350.4.c.b.649.1		2	40.3	even	4
1350.4.c.b.649.2		2	40.27	even	4
1350.4.c.s.649.1		2	120.107	odd	4
1350.4.c.s.649.2		2	120.83	odd	4
1728.4.a.k.1.1		1	1.1	even	1	trivial
1728.4.a.l.1.1		1	4.3	odd	2
1728.4.a.u.1.1		1	3.2	odd	2
1728.4.a.v.1.1		1	12.11	even	2