Newform orbit 7938.2.a.bz

• Label: 7938.2.a.bz
• Level: $7938$
• Weight: $2$
• Character orbit: 7938.a
• Self dual: yes
• Analytic conductor: $63.385$
• Analytic rank: $0$
• Dimension: $3$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(63.3852491245\)
Analytic rank:	\(0\)
Dimension:	\(3\)
Coefficient field:	3.3.321.1
Defining polynomial:	\( x^{3} - x^{2} - 4x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} + \beta_{2} q^{5} + q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - q^{5} + 3 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - \nu - 3 \)

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

0.239123
2.46050
−1.69963

1.00000

0

1.00000

−3.18194

0

0

1.00000

0

−3.18194

1.2

1.00000

0

1.00000

0.593579

0

0

1.00000

0

0.593579

1.3

1.00000

0

1.00000

1.58836

0

0

1.00000

0

1.58836

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	7938.2.a.bz		3
3.b	odd	2	1		7938.2.a.bw		3
7.b	odd	2	1		7938.2.a.ca		3
7.d	odd	6	2		1134.2.g.l		6
9.c	even	3	2		2646.2.f.m		6
9.d	odd	6	2		882.2.f.o		6
21.c	even	2	1		7938.2.a.bv		3
21.g	even	6	2		1134.2.g.m		6
63.g	even	3	2		2646.2.h.o		6
63.h	even	3	2		2646.2.e.p		6
63.i	even	6	2		126.2.e.c	✓	6
63.j	odd	6	2		882.2.e.o		6
63.k	odd	6	2		378.2.h.c		6
63.l	odd	6	2		2646.2.f.l		6
63.n	odd	6	2		882.2.h.p		6
63.o	even	6	2		882.2.f.n		6
63.s	even	6	2		126.2.h.d	yes	6
63.t	odd	6	2		378.2.e.d		6
252.n	even	6	2		3024.2.t.h		6
252.r	odd	6	2		1008.2.q.g		6
252.bj	even	6	2		3024.2.q.g		6
252.bn	odd	6	2		1008.2.t.h		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.2.e.c	✓	6	63.i	even	6	2
126.2.h.d	yes	6	63.s	even	6	2
378.2.e.d		6	63.t	odd	6	2
378.2.h.c		6	63.k	odd	6	2
882.2.e.o		6	63.j	odd	6	2
882.2.f.n		6	63.o	even	6	2
882.2.f.o		6	9.d	odd	6	2
882.2.h.p		6	63.n	odd	6	2
1008.2.q.g		6	252.r	odd	6	2
1008.2.t.h		6	252.bn	odd	6	2
1134.2.g.l		6	7.d	odd	6	2
1134.2.g.m		6	21.g	even	6	2
2646.2.e.p		6	63.h	even	3	2
2646.2.f.l		6	63.l	odd	6	2
2646.2.f.m		6	9.c	even	3	2
2646.2.h.o		6	63.g	even	3	2
3024.2.q.g		6	252.bj	even	6	2
3024.2.t.h		6	252.n	even	6	2
7938.2.a.bv		3	21.c	even	2	1
7938.2.a.bw		3	3.b	odd	2	1
7938.2.a.bz		3	1.a	even	1	1	trivial
7938.2.a.ca		3	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5}^{3} + T_{5}^{2} - 6T_{5} + 3 \)

\( T_{11}^{3} + T_{11}^{2} - 6T_{11} + 3 \)

\( T_{13}^{3} - 8T_{13}^{2} + T_{13} + 69 \)

\( T_{17}^{3} - 4T_{17}^{2} - 12T_{17} + 24 \)

\( T_{23}^{3} + 7T_{23}^{2} + 12T_{23} + 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{3} \)
$3$	\( T^{3} \)
$5$	\( T^{3} + T^{2} - 6T + 3 \)
$7$	\( T^{3} \)
$11$	\( T^{3} + T^{2} - 6T + 3 \)