Properties

Label 9408.2.a.dn
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
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: no (minimal twist has level 4704)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} + q^{9} -2 \beta q^{11} -3 \beta q^{13} -\beta q^{15} -3 \beta q^{17} + 8 q^{19} -2 \beta q^{23} -3 q^{25} - q^{27} -4 q^{31} + 2 \beta q^{33} + 4 q^{37} + 3 \beta q^{39} -5 \beta q^{41} -4 \beta q^{43} + \beta q^{45} + 4 q^{47} + 3 \beta q^{51} + 6 q^{53} -4 q^{55} -8 q^{57} + 8 q^{59} -3 \beta q^{61} -6 q^{65} + 2 \beta q^{69} + 10 \beta q^{71} + \beta q^{73} + 3 q^{75} -8 \beta q^{79} + q^{81} + 12 q^{83} -6 q^{85} + \beta q^{89} + 4 q^{93} + 8 \beta q^{95} + 9 \beta q^{97} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 16q^{19} - 6q^{25} - 2q^{27} - 8q^{31} + 8q^{37} + 8q^{47} + 12q^{53} - 8q^{55} - 16q^{57} + 16q^{59} - 12q^{65} + 6q^{75} + 2q^{81} + 24q^{83} - 12q^{85} + 8q^{93} + O(q^{100}) \)

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
0 −1.00000 0 −1.41421 0 0 0 1.00000 0
1.2 0 −1.00000 0 1.41421 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.dn 2
4.b odd 2 1 9408.2.a.dy 2
7.b odd 2 1 9408.2.a.dy 2
8.b even 2 1 4704.2.a.bo yes 2
8.d odd 2 1 4704.2.a.bl 2
28.d even 2 1 inner 9408.2.a.dn 2
56.e even 2 1 4704.2.a.bo yes 2
56.h odd 2 1 4704.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bl 2 8.d odd 2 1
4704.2.a.bl 2 56.h odd 2 1
4704.2.a.bo yes 2 8.b even 2 1
4704.2.a.bo yes 2 56.e even 2 1
9408.2.a.dn 2 1.a even 1 1 trivial
9408.2.a.dn 2 28.d even 2 1 inner
9408.2.a.dy 2 4.b odd 2 1
9408.2.a.dy 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} - 2 \)
\( T_{11}^{2} - 8 \)
\( T_{13}^{2} - 18 \)
\( T_{17}^{2} - 18 \)
\( T_{19} - 8 \)
\( T_{31} + 4 \)