Properties

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

Related objects

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{5} + q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{5} + q^{9} - 4q^{11} - 6q^{13} + 2q^{15} + 2q^{17} - 4q^{19} + 4q^{23} - q^{25} - q^{27} + 2q^{29} + 8q^{31} + 4q^{33} + 10q^{37} + 6q^{39} + 2q^{41} + 8q^{43} - 2q^{45} - 2q^{51} + 10q^{53} + 8q^{55} + 4q^{57} + 12q^{59} + 10q^{61} + 12q^{65} - 8q^{67} - 4q^{69} - 12q^{71} - 2q^{73} + q^{75} + q^{81} - 12q^{83} - 4q^{85} - 2q^{87} - 6q^{89} - 8q^{93} + 8q^{95} - 2q^{97} - 4q^{99} + O(q^{100}) \)

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:

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.g 1
4.b odd 2 1 9408.2.a.cb 1
7.b odd 2 1 1344.2.a.r 1
8.b even 2 1 4704.2.a.be 1
8.d odd 2 1 4704.2.a.m 1
21.c even 2 1 4032.2.a.n 1
28.d even 2 1 1344.2.a.h 1
56.e even 2 1 672.2.a.f yes 1
56.h odd 2 1 672.2.a.b 1
84.h odd 2 1 4032.2.a.f 1
112.j even 4 2 5376.2.c.s 2
112.l odd 4 2 5376.2.c.m 2
168.e odd 2 1 2016.2.a.k 1
168.i even 2 1 2016.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.b 1 56.h odd 2 1
672.2.a.f yes 1 56.e even 2 1
1344.2.a.h 1 28.d even 2 1
1344.2.a.r 1 7.b odd 2 1
2016.2.a.k 1 168.e odd 2 1
2016.2.a.l 1 168.i even 2 1
4032.2.a.f 1 84.h odd 2 1
4032.2.a.n 1 21.c even 2 1
4704.2.a.m 1 8.d odd 2 1
4704.2.a.be 1 8.b even 2 1
5376.2.c.m 2 112.l odd 4 2
5376.2.c.s 2 112.j even 4 2
9408.2.a.g 1 1.a even 1 1 trivial
9408.2.a.cb 1 4.b odd 2 1

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

\( T_{5} + 2 \)
\( T_{11} + 4 \)
\( T_{13} + 6 \)
\( T_{17} - 2 \)
\( T_{19} + 4 \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 - 10 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 10 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 - 10 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 + 12 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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