Properties

Label 9408.2.a.cs
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 2 q^{5} + q^{9} - 6 q^{11} - 3 q^{13} + 2 q^{15} - 4 q^{17} + 5 q^{19} + 4 q^{23} - q^{25} + q^{27} + 4 q^{29} + 7 q^{31} - 6 q^{33} + 9 q^{37} - 3 q^{39} + 2 q^{41} - q^{43} + 2 q^{45} + 2 q^{47} - 4 q^{51} - 8 q^{53} - 12 q^{55} + 5 q^{57} + 10 q^{61} - 6 q^{65} - 15 q^{67} + 4 q^{69} + 6 q^{71} + 11 q^{73} - q^{75} - q^{79} + q^{81} - 6 q^{83} - 8 q^{85} + 4 q^{87} + 8 q^{89} + 7 q^{93} + 10 q^{95} + 14 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0
0 1.00000 0 2.00000 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

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 9408.2.a.cs 1
4.b odd 2 1 9408.2.a.bk 1
7.b odd 2 1 9408.2.a.f 1
7.d odd 6 2 1344.2.q.t 2
8.b even 2 1 2352.2.a.e 1
8.d odd 2 1 1176.2.a.e 1
24.f even 2 1 3528.2.a.y 1
24.h odd 2 1 7056.2.a.bn 1
28.d even 2 1 9408.2.a.cd 1
28.f even 6 2 1344.2.q.i 2
56.e even 2 1 1176.2.a.d 1
56.h odd 2 1 2352.2.a.x 1
56.j odd 6 2 336.2.q.a 2
56.k odd 6 2 1176.2.q.e 2
56.m even 6 2 168.2.q.b 2
56.p even 6 2 2352.2.q.v 2
168.e odd 2 1 3528.2.a.f 1
168.i even 2 1 7056.2.a.i 1
168.v even 6 2 3528.2.s.d 2
168.ba even 6 2 1008.2.s.m 2
168.be odd 6 2 504.2.s.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 56.m even 6 2
336.2.q.a 2 56.j odd 6 2
504.2.s.g 2 168.be odd 6 2
1008.2.s.m 2 168.ba even 6 2
1176.2.a.d 1 56.e even 2 1
1176.2.a.e 1 8.d odd 2 1
1176.2.q.e 2 56.k odd 6 2
1344.2.q.i 2 28.f even 6 2
1344.2.q.t 2 7.d odd 6 2
2352.2.a.e 1 8.b even 2 1
2352.2.a.x 1 56.h odd 2 1
2352.2.q.v 2 56.p even 6 2
3528.2.a.f 1 168.e odd 2 1
3528.2.a.y 1 24.f even 2 1
3528.2.s.d 2 168.v even 6 2
7056.2.a.i 1 168.i even 2 1
7056.2.a.bn 1 24.h odd 2 1
9408.2.a.f 1 7.b odd 2 1
9408.2.a.bk 1 4.b odd 2 1
9408.2.a.cd 1 28.d even 2 1
9408.2.a.cs 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display
\( T_{13} + 3 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{19} - 5 \) Copy content Toggle raw display
\( T_{31} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T + 3 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 4 \) Copy content Toggle raw display
$31$ \( T - 7 \) Copy content Toggle raw display
$37$ \( T - 9 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T + 8 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 15 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 8 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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