Newform orbit 4998.2.a.bs

• Label: 4998.2.a.bs
• Level: $4998$
• Weight: $2$
• Character orbit: 4998.a
• Self dual: yes
• Analytic conductor: $39.909$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(39.9092309302\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{41}) \)
Defining polynomial:	\( x^{2} - x - 10 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 714)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} - q^{3} + q^{4} - \beta q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \)

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

3.70156
−2.70156

−1.00000

−1.00000

1.00000

−3.70156

1.00000

0

−1.00000

1.00000

3.70156

1.2

−1.00000

−1.00000

1.00000

2.70156

1.00000

0

−1.00000

1.00000

−2.70156

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(-1\)
\(17\)	\(-1\)

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	4998.2.a.bs		2
7.b	odd	2	1		714.2.a.k	✓	2
21.c	even	2	1		2142.2.a.y		2
28.d	even	2	1		5712.2.a.bi		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
714.2.a.k	✓	2	7.b	odd	2	1
2142.2.a.y		2	21.c	even	2	1
4998.2.a.bs		2	1.a	even	1	1	trivial
5712.2.a.bi		2	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5}^{2} + T_{5} - 10 \)

\( T_{11}^{2} - T_{11} - 10 \)

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

\( T_{23}^{2} - 6T_{23} - 32 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} + T - 10 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - T - 10 \)