Properties

Label 9408.2.a.ea
Level 9408
Weight 2
Character orbit 9408.a
Self dual yes
Analytic conductor 75.123
Analytic rank 1
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: \(1\)
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: 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.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} + q^{9} -2 \beta q^{11} + \beta q^{13} + \beta q^{15} + \beta q^{17} -2 \beta q^{23} -3 q^{25} + q^{27} -4 q^{31} -2 \beta q^{33} -4 q^{37} + \beta q^{39} -\beta q^{41} -4 \beta q^{43} + \beta q^{45} -12 q^{47} + \beta q^{51} -10 q^{53} -4 q^{55} + \beta q^{61} + 2 q^{65} -8 \beta q^{67} -2 \beta q^{69} + 2 \beta q^{71} + 9 \beta q^{73} -3 q^{75} -8 \beta q^{79} + q^{81} + 4 q^{83} + 2 q^{85} + 5 \beta q^{89} -4 q^{93} -7 \beta q^{97} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{9} - 6q^{25} + 2q^{27} - 8q^{31} - 8q^{37} - 24q^{47} - 20q^{53} - 8q^{55} + 4q^{65} - 6q^{75} + 2q^{81} + 8q^{83} + 4q^{85} - 8q^{93} + O(q^{100}) \)

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:

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.ea 2
4.b odd 2 1 9408.2.a.dl 2
7.b odd 2 1 9408.2.a.dl 2
8.b even 2 1 4704.2.a.bj 2
8.d odd 2 1 4704.2.a.bq yes 2
28.d even 2 1 inner 9408.2.a.ea 2
56.e even 2 1 4704.2.a.bj 2
56.h odd 2 1 4704.2.a.bq yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bj 2 8.b even 2 1
4704.2.a.bj 2 56.e even 2 1
4704.2.a.bq yes 2 8.d odd 2 1
4704.2.a.bq yes 2 56.h odd 2 1
9408.2.a.dl 2 4.b odd 2 1
9408.2.a.dl 2 7.b odd 2 1
9408.2.a.ea 2 1.a even 1 1 trivial
9408.2.a.ea 2 28.d even 2 1 inner

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + 14 T^{2} + 121 T^{4} \)
$13$ \( 1 + 24 T^{2} + 169 T^{4} \)
$17$ \( 1 + 32 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 38 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 80 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 54 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 12 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 + 10 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( 1 + 120 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 6 T^{2} + 4489 T^{4} \)
$71$ \( 1 + 134 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 16 T^{2} + 5329 T^{4} \)
$79$ \( 1 + 30 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 128 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 96 T^{2} + 9409 T^{4} \)
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