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

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

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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} - 2 \)
\( T_{11} + 6 \)
\( T_{13} + 3 \)
\( T_{17} + 4 \)
\( T_{19} - 5 \)
\( T_{31} - 7 \)

Hecke characteristic polynomials

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