Newform orbit 1728.3.e.a

• Label: 1728.3.e.a
• Level: $1728$
• Weight: $3$
• Character orbit: 1728.e
• Self dual: yes
• Analytic conductor: $47.085$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(47.0845896815\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 13 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(703\)	\(1217\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1025.1

0

0

0

0

0

0

−13.0000

0

0

0

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	1728.3.e.a		1
3.b	odd	2	1	CM	1728.3.e.a		1
4.b	odd	2	1		1728.3.e.d		1
8.b	even	2	1		27.3.b.a	✓	1
8.d	odd	2	1		432.3.e.b		1
12.b	even	2	1		1728.3.e.d		1
24.f	even	2	1		432.3.e.b		1
24.h	odd	2	1		27.3.b.a	✓	1
40.f	even	2	1		675.3.c.c		1
40.i	odd	4	2		675.3.d.c		2
72.j	odd	6	2		81.3.d.a		2
72.l	even	6	2		1296.3.q.a		2
72.n	even	6	2		81.3.d.a		2
72.p	odd	6	2		1296.3.q.a		2
120.i	odd	2	1		675.3.c.c		1
120.w	even	4	2		675.3.d.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.3.b.a	✓	1	8.b	even	2	1
27.3.b.a	✓	1	24.h	odd	2	1
81.3.d.a		2	72.j	odd	6	2
81.3.d.a		2	72.n	even	6	2
432.3.e.b		1	8.d	odd	2	1
432.3.e.b		1	24.f	even	2	1
675.3.c.c		1	40.f	even	2	1
675.3.c.c		1	120.i	odd	2	1
675.3.d.c		2	40.i	odd	4	2
675.3.d.c		2	120.w	even	4	2
1296.3.q.a		2	72.l	even	6	2
1296.3.q.a		2	72.p	odd	6	2
1728.3.e.a		1	1.a	even	1	1	trivial
1728.3.e.a		1	3.b	odd	2	1	CM
1728.3.e.d		1	4.b	odd	2	1
1728.3.e.d		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5} \)

\( T_{7} + 13 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T + 13 \)
$11$	\( T \)