Newform orbit 1728.2.a.o

• Label: 1728.2.a.o
• Level: $1728$
• Weight: $2$
• Character orbit: 1728.a
• Self dual: yes
• Analytic conductor: $13.798$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	\(1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{7}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

0

0

0

0

0

1.00000

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(-1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1728.2.a.o		1
3.b	odd	2	1	CM	1728.2.a.o		1
4.b	odd	2	1		1728.2.a.n		1
8.b	even	2	1		432.2.a.e		1
8.d	odd	2	1		27.2.a.a	✓	1
12.b	even	2	1		1728.2.a.n		1
24.f	even	2	1		27.2.a.a	✓	1
24.h	odd	2	1		432.2.a.e		1
40.e	odd	2	1		675.2.a.e		1
40.k	even	4	2		675.2.b.f		2
56.e	even	2	1		1323.2.a.i		1
72.j	odd	6	2		1296.2.i.i		2
72.l	even	6	2		81.2.c.a		2
72.n	even	6	2		1296.2.i.i		2
72.p	odd	6	2		81.2.c.a		2
88.g	even	2	1		3267.2.a.f		1
104.h	odd	2	1		4563.2.a.e		1
120.m	even	2	1		675.2.a.e		1
120.q	odd	4	2		675.2.b.f		2
136.e	odd	2	1		7803.2.a.k		1
152.b	even	2	1		9747.2.a.f		1
168.e	odd	2	1		1323.2.a.i		1
216.r	odd	18	6		729.2.e.f		6
216.v	even	18	6		729.2.e.f		6
264.p	odd	2	1		3267.2.a.f		1
312.h	even	2	1		4563.2.a.e		1
408.h	even	2	1		7803.2.a.k		1
456.l	odd	2	1		9747.2.a.f		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.2.a.a	✓	1	8.d	odd	2	1
27.2.a.a	✓	1	24.f	even	2	1
81.2.c.a		2	72.l	even	6	2
81.2.c.a		2	72.p	odd	6	2
432.2.a.e		1	8.b	even	2	1
432.2.a.e		1	24.h	odd	2	1
675.2.a.e		1	40.e	odd	2	1
675.2.a.e		1	120.m	even	2	1
675.2.b.f		2	40.k	even	4	2
675.2.b.f		2	120.q	odd	4	2
729.2.e.f		6	216.r	odd	18	6
729.2.e.f		6	216.v	even	18	6
1296.2.i.i		2	72.j	odd	6	2
1296.2.i.i		2	72.n	even	6	2
1323.2.a.i		1	56.e	even	2	1
1323.2.a.i		1	168.e	odd	2	1
1728.2.a.n		1	4.b	odd	2	1
1728.2.a.n		1	12.b	even	2	1
1728.2.a.o		1	1.a	even	1	1	trivial
1728.2.a.o		1	3.b	odd	2	1	CM
3267.2.a.f		1	88.g	even	2	1
3267.2.a.f		1	264.p	odd	2	1
4563.2.a.e		1	104.h	odd	2	1
4563.2.a.e		1	312.h	even	2	1
7803.2.a.k		1	136.e	odd	2	1
7803.2.a.k		1	408.h	even	2	1
9747.2.a.f		1	152.b	even	2	1
9747.2.a.f		1	456.l	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5} \)

\( T_{7} - 1 \)

\( T_{11} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T - 1 \)
$11$	\( T \)