Properties

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

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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 21)
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 \( q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} - q^{25} - q^{27} + 2 q^{29} - 4 q^{33} - 6 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} - 6 q^{51} - 6 q^{53} - 8 q^{55} + 4 q^{57} - 12 q^{59} - 2 q^{61} + 4 q^{65} + 4 q^{67} + 6 q^{73} + q^{75} + 16 q^{79} + q^{81} + 12 q^{83} - 12 q^{85} - 2 q^{87} + 14 q^{89} + 8 q^{95} - 18 q^{97} + 4 q^{99}+O(q^{100}) \) 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.m 1
4.b odd 2 1 9408.2.a.bv 1
7.b odd 2 1 1344.2.a.s 1
8.b even 2 1 2352.2.a.v 1
8.d odd 2 1 147.2.a.a 1
21.c even 2 1 4032.2.a.k 1
24.f even 2 1 441.2.a.f 1
24.h odd 2 1 7056.2.a.p 1
28.d even 2 1 1344.2.a.g 1
40.e odd 2 1 3675.2.a.n 1
56.e even 2 1 21.2.a.a 1
56.h odd 2 1 336.2.a.a 1
56.j odd 6 2 2352.2.q.x 2
56.k odd 6 2 147.2.e.c 2
56.m even 6 2 147.2.e.b 2
56.p even 6 2 2352.2.q.e 2
84.h odd 2 1 4032.2.a.h 1
112.j even 4 2 5376.2.c.r 2
112.l odd 4 2 5376.2.c.l 2
168.e odd 2 1 63.2.a.a 1
168.i even 2 1 1008.2.a.l 1
168.v even 6 2 441.2.e.b 2
168.be odd 6 2 441.2.e.a 2
280.c odd 2 1 8400.2.a.bn 1
280.n even 2 1 525.2.a.d 1
280.y odd 4 2 525.2.d.a 2
504.be even 6 2 567.2.f.g 2
504.co odd 6 2 567.2.f.b 2
616.g odd 2 1 2541.2.a.j 1
728.b even 2 1 3549.2.a.c 1
840.b odd 2 1 1575.2.a.c 1
840.bm even 4 2 1575.2.d.a 2
952.k even 2 1 6069.2.a.b 1
1064.p odd 2 1 7581.2.a.d 1
1848.e even 2 1 7623.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 56.e even 2 1
63.2.a.a 1 168.e odd 2 1
147.2.a.a 1 8.d odd 2 1
147.2.e.b 2 56.m even 6 2
147.2.e.c 2 56.k odd 6 2
336.2.a.a 1 56.h odd 2 1
441.2.a.f 1 24.f even 2 1
441.2.e.a 2 168.be odd 6 2
441.2.e.b 2 168.v even 6 2
525.2.a.d 1 280.n even 2 1
525.2.d.a 2 280.y odd 4 2
567.2.f.b 2 504.co odd 6 2
567.2.f.g 2 504.be even 6 2
1008.2.a.l 1 168.i even 2 1
1344.2.a.g 1 28.d even 2 1
1344.2.a.s 1 7.b odd 2 1
1575.2.a.c 1 840.b odd 2 1
1575.2.d.a 2 840.bm even 4 2
2352.2.a.v 1 8.b even 2 1
2352.2.q.e 2 56.p even 6 2
2352.2.q.x 2 56.j odd 6 2
2541.2.a.j 1 616.g odd 2 1
3549.2.a.c 1 728.b even 2 1
3675.2.a.n 1 40.e odd 2 1
4032.2.a.h 1 84.h odd 2 1
4032.2.a.k 1 21.c even 2 1
5376.2.c.l 2 112.l odd 4 2
5376.2.c.r 2 112.j even 4 2
6069.2.a.b 1 952.k even 2 1
7056.2.a.p 1 24.h odd 2 1
7581.2.a.d 1 1064.p odd 2 1
7623.2.a.g 1 1848.e even 2 1
8400.2.a.bn 1 280.c odd 2 1
9408.2.a.m 1 1.a even 1 1 trivial
9408.2.a.bv 1 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display
\( T_{31} \) 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
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T + 18 \) Copy content Toggle raw display
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