Newform orbit 7488.2.a.w

• Label: 7488.2.a.w
• Level: $7488$
• Weight: $2$
• Character orbit: 7488.a
• Self dual: yes
• Analytic conductor: $59.792$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(59.7919810335\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

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

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

−1.00000

0

1.00000

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)
\(13\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7488.2.a.w		1
3.b	odd	2	1		832.2.a.j		1
4.b	odd	2	1		7488.2.a.v		1
8.b	even	2	1		234.2.a.b		1
8.d	odd	2	1		1872.2.a.m		1
12.b	even	2	1		832.2.a.a		1
24.f	even	2	1		208.2.a.d		1
24.h	odd	2	1		26.2.a.b	✓	1
40.f	even	2	1		5850.2.a.bn		1
40.i	odd	4	2		5850.2.e.v		2
48.i	odd	4	2		3328.2.b.g		2
48.k	even	4	2		3328.2.b.k		2
72.j	odd	6	2		2106.2.e.h		2
72.n	even	6	2		2106.2.e.t		2
104.e	even	2	1		3042.2.a.l		1
104.j	odd	4	2		3042.2.b.f		2
120.i	odd	2	1		650.2.a.g		1
120.m	even	2	1		5200.2.a.c		1
120.w	even	4	2		650.2.b.a		2
168.i	even	2	1		1274.2.a.o		1
168.s	odd	6	2		1274.2.f.l		2
168.ba	even	6	2		1274.2.f.a		2
264.m	even	2	1		3146.2.a.a		1
312.b	odd	2	1		338.2.a.a		1
312.h	even	2	1		2704.2.a.n		1
312.w	odd	4	2		2704.2.f.j		2
312.y	even	4	2		338.2.b.a		2
312.bg	odd	6	2		338.2.c.g		2
312.bh	odd	6	2		338.2.c.c		2
312.bo	even	12	4		338.2.e.d		4
408.b	odd	2	1		7514.2.a.i		1
456.p	even	2	1		9386.2.a.f		1
1560.y	odd	2	1		8450.2.a.y		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.2.a.b	✓	1	24.h	odd	2	1
208.2.a.d		1	24.f	even	2	1
234.2.a.b		1	8.b	even	2	1
338.2.a.a		1	312.b	odd	2	1
338.2.b.a		2	312.y	even	4	2
338.2.c.c		2	312.bh	odd	6	2
338.2.c.g		2	312.bg	odd	6	2
338.2.e.d		4	312.bo	even	12	4
650.2.a.g		1	120.i	odd	2	1
650.2.b.a		2	120.w	even	4	2
832.2.a.a		1	12.b	even	2	1
832.2.a.j		1	3.b	odd	2	1
1274.2.a.o		1	168.i	even	2	1
1274.2.f.a		2	168.ba	even	6	2
1274.2.f.l		2	168.s	odd	6	2
1872.2.a.m		1	8.d	odd	2	1
2106.2.e.h		2	72.j	odd	6	2
2106.2.e.t		2	72.n	even	6	2
2704.2.a.n		1	312.h	even	2	1
2704.2.f.j		2	312.w	odd	4	2
3042.2.a.l		1	104.e	even	2	1
3042.2.b.f		2	104.j	odd	4	2
3146.2.a.a		1	264.m	even	2	1
3328.2.b.g		2	48.i	odd	4	2
3328.2.b.k		2	48.k	even	4	2
5200.2.a.c		1	120.m	even	2	1
5850.2.a.bn		1	40.f	even	2	1
5850.2.e.v		2	40.i	odd	4	2
7488.2.a.v		1	4.b	odd	2	1
7488.2.a.w		1	1.a	even	1	1	trivial
7514.2.a.i		1	408.b	odd	2	1
8450.2.a.y		1	1560.y	odd	2	1
9386.2.a.f		1	456.p	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7488))\):

\( T_{5} + 1 \)

\( T_{7} - 1 \)

\( T_{11} + 2 \)

\( T_{17} - 3 \)

\( T_{19} + 6 \)

\( T_{29} - 2 \)

Hecke characteristic polynomials

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