Newform orbit 48.3.e.a

• Label: 48.3.e.a
• Level: $48$
• Weight: $3$
• Character orbit: 48.e
• Self dual: yes
• Analytic conductor: $1.308$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.30790526893\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10}) \)

Expression as an eta quotient

\(f(z) = \dfrac{\eta(4z)^{9}\eta(12z)^{9}}{\eta(2z)^{3}\eta(6z)^{3}\eta(8z)^{3}\eta(24z)^{3}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-3}(1 - q^{4n})^{9}(1 - q^{6n})^{-3}(1 - q^{8n})^{-3}(1 - q^{12n})^{9}(1 - q^{24n})^{-3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

17.1

0

0

3.00000

0

0

0

−2.00000

0

9.00000

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	48.3.e.a		1
3.b	odd	2	1	CM	48.3.e.a		1
4.b	odd	2	1		12.3.c.a	✓	1
5.b	even	2	1		1200.3.l.b		1
5.c	odd	4	2		1200.3.c.c		2
8.b	even	2	1		192.3.e.a		1
8.d	odd	2	1		192.3.e.b		1
9.c	even	3	2		1296.3.q.b		2
9.d	odd	6	2		1296.3.q.b		2
12.b	even	2	1		12.3.c.a	✓	1
15.d	odd	2	1		1200.3.l.b		1
15.e	even	4	2		1200.3.c.c		2
16.e	even	4	2		768.3.h.b		2
16.f	odd	4	2		768.3.h.a		2
20.d	odd	2	1		300.3.g.b		1
20.e	even	4	2		300.3.b.a		2
24.f	even	2	1		192.3.e.b		1
24.h	odd	2	1		192.3.e.a		1
28.d	even	2	1		588.3.c.c		1
28.f	even	6	2		588.3.p.b		2
28.g	odd	6	2		588.3.p.c		2
36.f	odd	6	2		324.3.g.b		2
36.h	even	6	2		324.3.g.b		2
44.c	even	2	1		1452.3.e.b		1
48.i	odd	4	2		768.3.h.b		2
48.k	even	4	2		768.3.h.a		2
60.h	even	2	1		300.3.g.b		1
60.l	odd	4	2		300.3.b.a		2
84.h	odd	2	1		588.3.c.c		1
84.j	odd	6	2		588.3.p.b		2
84.n	even	6	2		588.3.p.c		2
132.d	odd	2	1		1452.3.e.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.3.c.a	✓	1	4.b	odd	2	1
12.3.c.a	✓	1	12.b	even	2	1
48.3.e.a		1	1.a	even	1	1	trivial
48.3.e.a		1	3.b	odd	2	1	CM
192.3.e.a		1	8.b	even	2	1
192.3.e.a		1	24.h	odd	2	1
192.3.e.b		1	8.d	odd	2	1
192.3.e.b		1	24.f	even	2	1
300.3.b.a		2	20.e	even	4	2
300.3.b.a		2	60.l	odd	4	2
300.3.g.b		1	20.d	odd	2	1
300.3.g.b		1	60.h	even	2	1
324.3.g.b		2	36.f	odd	6	2
324.3.g.b		2	36.h	even	6	2
588.3.c.c		1	28.d	even	2	1
588.3.c.c		1	84.h	odd	2	1
588.3.p.b		2	28.f	even	6	2
588.3.p.b		2	84.j	odd	6	2
588.3.p.c		2	28.g	odd	6	2
588.3.p.c		2	84.n	even	6	2
768.3.h.a		2	16.f	odd	4	2
768.3.h.a		2	48.k	even	4	2
768.3.h.b		2	16.e	even	4	2
768.3.h.b		2	48.i	odd	4	2
1200.3.c.c		2	5.c	odd	4	2
1200.3.c.c		2	15.e	even	4	2
1200.3.l.b		1	5.b	even	2	1
1200.3.l.b		1	15.d	odd	2	1
1296.3.q.b		2	9.c	even	3	2
1296.3.q.b		2	9.d	odd	6	2
1452.3.e.b		1	44.c	even	2	1
1452.3.e.b		1	132.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

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