Newform orbit 4032.2.a.r

• Label: 4032.2.a.r
• Level: $4032$
• Weight: $2$
• Character orbit: 4032.a
• Self dual: yes
• Analytic conductor: $32.196$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.1956820950\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(7\)	\( +1 \)

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	4032.2.a.r		1
3.b	odd	2	1		448.2.a.a		1
4.b	odd	2	1		4032.2.a.w		1
8.b	even	2	1		1008.2.a.h		1
8.d	odd	2	1		126.2.a.b		1
12.b	even	2	1		448.2.a.g		1
21.c	even	2	1		3136.2.a.z		1
24.f	even	2	1		14.2.a.a	✓	1
24.h	odd	2	1		112.2.a.c		1
40.e	odd	2	1		3150.2.a.i		1
40.k	even	4	2		3150.2.g.j		2
48.i	odd	4	2		1792.2.b.g		2
48.k	even	4	2		1792.2.b.c		2
56.e	even	2	1		882.2.a.i		1
56.h	odd	2	1		7056.2.a.bd		1
56.k	odd	6	2		882.2.g.c		2
56.m	even	6	2		882.2.g.d		2
72.l	even	6	2		1134.2.f.l		2
72.p	odd	6	2		1134.2.f.f		2
84.h	odd	2	1		3136.2.a.e		1
120.i	odd	2	1		2800.2.a.g		1
120.m	even	2	1		350.2.a.f		1
120.q	odd	4	2		350.2.c.d		2
120.w	even	4	2		2800.2.g.h		2
168.e	odd	2	1		98.2.a.a		1
168.i	even	2	1		784.2.a.b		1
168.s	odd	6	2		784.2.i.c		2
168.v	even	6	2		98.2.c.b		2
168.ba	even	6	2		784.2.i.i		2
168.be	odd	6	2		98.2.c.a		2
264.p	odd	2	1		1694.2.a.e		1
312.h	even	2	1		2366.2.a.j		1
312.w	odd	4	2		2366.2.d.b		2
408.h	even	2	1		4046.2.a.f		1
456.l	odd	2	1		5054.2.a.c		1
552.h	odd	2	1		7406.2.a.a		1
840.b	odd	2	1		2450.2.a.t		1
840.bm	even	4	2		2450.2.c.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.2.a.a	✓	1	24.f	even	2	1
98.2.a.a		1	168.e	odd	2	1
98.2.c.a		2	168.be	odd	6	2
98.2.c.b		2	168.v	even	6	2
112.2.a.c		1	24.h	odd	2	1
126.2.a.b		1	8.d	odd	2	1
350.2.a.f		1	120.m	even	2	1
350.2.c.d		2	120.q	odd	4	2
448.2.a.a		1	3.b	odd	2	1
448.2.a.g		1	12.b	even	2	1
784.2.a.b		1	168.i	even	2	1
784.2.i.c		2	168.s	odd	6	2
784.2.i.i		2	168.ba	even	6	2
882.2.a.i		1	56.e	even	2	1
882.2.g.c		2	56.k	odd	6	2
882.2.g.d		2	56.m	even	6	2
1008.2.a.h		1	8.b	even	2	1
1134.2.f.f		2	72.p	odd	6	2
1134.2.f.l		2	72.l	even	6	2
1694.2.a.e		1	264.p	odd	2	1
1792.2.b.c		2	48.k	even	4	2
1792.2.b.g		2	48.i	odd	4	2
2366.2.a.j		1	312.h	even	2	1
2366.2.d.b		2	312.w	odd	4	2
2450.2.a.t		1	840.b	odd	2	1
2450.2.c.c		2	840.bm	even	4	2
2800.2.a.g		1	120.i	odd	2	1
2800.2.g.h		2	120.w	even	4	2
3136.2.a.e		1	84.h	odd	2	1
3136.2.a.z		1	21.c	even	2	1
3150.2.a.i		1	40.e	odd	2	1
3150.2.g.j		2	40.k	even	4	2
4032.2.a.r		1	1.a	even	1	1	trivial
4032.2.a.w		1	4.b	odd	2	1
4046.2.a.f		1	408.h	even	2	1
5054.2.a.c		1	456.l	odd	2	1
7056.2.a.bd		1	56.h	odd	2	1
7406.2.a.a		1	552.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\):

\( T_{5} \)

\( T_{11} \)

\( T_{13} - 4 \)

\( T_{17} + 6 \)

\( T_{19} - 2 \)

Hecke characteristic polynomials

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