Newform orbit 48.7.e.a

• Label: 48.7.e.a
• Level: $48$
• Weight: $7$
• Character orbit: 48.e
• Self dual: yes
• Analytic conductor: $11.043$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.0425960138\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 27 q^{3} + 286 q^{7} + 729 q^{9}+O(q^{10}) \)

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

27.0000

0

0

0

286.000

0

729.000

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	48.7.e.a		1
3.b	odd	2	1	CM	48.7.e.a		1
4.b	odd	2	1		3.7.b.a	✓	1
8.b	even	2	1		192.7.e.a		1
8.d	odd	2	1		192.7.e.b		1
12.b	even	2	1		3.7.b.a	✓	1
20.d	odd	2	1		75.7.c.a		1
20.e	even	4	2		75.7.d.a		2
24.f	even	2	1		192.7.e.b		1
24.h	odd	2	1		192.7.e.a		1
28.d	even	2	1		147.7.b.a		1
36.f	odd	6	2		81.7.d.a		2
36.h	even	6	2		81.7.d.a		2
60.h	even	2	1		75.7.c.a		1
60.l	odd	4	2		75.7.d.a		2
84.h	odd	2	1		147.7.b.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.7.b.a	✓	1	4.b	odd	2	1
3.7.b.a	✓	1	12.b	even	2	1
48.7.e.a		1	1.a	even	1	1	trivial
48.7.e.a		1	3.b	odd	2	1	CM
75.7.c.a		1	20.d	odd	2	1
75.7.c.a		1	60.h	even	2	1
75.7.d.a		2	20.e	even	4	2
75.7.d.a		2	60.l	odd	4	2
81.7.d.a		2	36.f	odd	6	2
81.7.d.a		2	36.h	even	6	2
147.7.b.a		1	28.d	even	2	1
147.7.b.a		1	84.h	odd	2	1
192.7.e.a		1	8.b	even	2	1
192.7.e.a		1	24.h	odd	2	1
192.7.e.b		1	8.d	odd	2	1
192.7.e.b		1	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{7}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T - 27 \)
$5$	\( T \)
$7$	\( T - 286 \)
$11$	\( T \)