Newform orbit 9702.2.a.dg

• Label: 9702.2.a.dg
• Level: $9702$
• Weight: $2$
• Character orbit: 9702.a
• Self dual: yes
• Analytic conductor: $77.471$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(77.4708600410\)
Analytic rank:	\(1\)
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.

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

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

1.00000

0

1.00000

−1.41421

0

0

1.00000

0

−1.41421

1.2

1.00000

0

1.00000

1.41421

0

0

1.00000

0

1.41421

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)
\(7\)	\(-1\)
\(11\)	\(1\)

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	9702.2.a.dg	yes	2
3.b	odd	2	1		9702.2.a.ct	✓	2
7.b	odd	2	1	inner	9702.2.a.dg	yes	2
21.c	even	2	1		9702.2.a.ct	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9702.2.a.ct	✓	2	3.b	odd	2	1
9702.2.a.ct	✓	2	21.c	even	2	1
9702.2.a.dg	yes	2	1.a	even	1	1	trivial
9702.2.a.dg	yes	2	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

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

\( T_{13}^{2} - 32 \)

\( T_{17}^{2} - 2 \)

\( T_{19}^{2} - 18 \)

\( T_{23} + 4 \)

\( T_{29} \)

Hecke characteristic polynomials

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