Newform orbit 98.4.a.g

• Label: 98.4.a.g
• Level: $98$
• Weight: $4$
• Character orbit: 98.a
• Self dual: yes
• Analytic conductor: $5.782$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.78218718056\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\(x^{2} - 2\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q -2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 14 \beta q^{5} -10 \beta q^{6} -8 q^{8} + 23 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 46q^{9} + O(q^{10}) \)

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

−1.41421
1.41421

−2.00000

−7.07107

4.00000

−19.7990

14.1421

0

−8.00000

23.0000

39.5980

1.2

−2.00000

7.07107

4.00000

19.7990

−14.1421

0

−8.00000

23.0000

−39.5980

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(7\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	98.4.a.g	✓	2
3.b	odd	2	1		882.4.a.bg		2
4.b	odd	2	1		784.4.a.y		2
5.b	even	2	1		2450.4.a.bx		2
7.b	odd	2	1	inner	98.4.a.g	✓	2
7.c	even	3	2		98.4.c.h		4
7.d	odd	6	2		98.4.c.h		4
21.c	even	2	1		882.4.a.bg		2
21.g	even	6	2		882.4.g.ba		4
21.h	odd	6	2		882.4.g.ba		4
28.d	even	2	1		784.4.a.y		2
35.c	odd	2	1		2450.4.a.bx		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
98.4.a.g	✓	2	1.a	even	1	1	trivial
98.4.a.g	✓	2	7.b	odd	2	1	inner
98.4.c.h		4	7.c	even	3	2
98.4.c.h		4	7.d	odd	6	2
784.4.a.y		2	4.b	odd	2	1
784.4.a.y		2	28.d	even	2	1
882.4.a.bg		2	3.b	odd	2	1
882.4.a.bg		2	21.c	even	2	1
882.4.g.ba		4	21.g	even	6	2
882.4.g.ba		4	21.h	odd	6	2
2450.4.a.bx		2	5.b	even	2	1
2450.4.a.bx		2	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 50 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( ( 2 + T )^{2} \)
$3$	\( -50 + T^{2} \)
$5$	\( -392 + T^{2} \)
$7$	\( T^{2} \)
$11$	\( ( 14 + T )^{2} \)