Properties

Label 9408.2.a.dc
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 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 3q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + 3q^{5} + q^{9} + q^{11} - 4q^{13} + 3q^{15} - 4q^{17} + 8q^{23} + 4q^{25} + q^{27} + 7q^{29} - 11q^{31} + q^{33} - 4q^{37} - 4q^{39} + 4q^{41} + 2q^{43} + 3q^{45} + 2q^{47} - 4q^{51} + 11q^{53} + 3q^{55} + 7q^{59} + 10q^{61} - 12q^{65} - 10q^{67} + 8q^{69} + 6q^{71} + 6q^{73} + 4q^{75} + 11q^{79} + q^{81} + 11q^{83} - 12q^{85} + 7q^{87} - 6q^{89} - 11q^{93} - 7q^{97} + q^{99} + O(q^{100}) \)

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 3.00000 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.dc 1
4.b odd 2 1 9408.2.a.bl 1
7.b odd 2 1 9408.2.a.e 1
7.d odd 6 2 1344.2.q.u 2
8.b even 2 1 4704.2.a.b 1
8.d odd 2 1 4704.2.a.s 1
28.d even 2 1 9408.2.a.bt 1
28.f even 6 2 1344.2.q.k 2
56.e even 2 1 4704.2.a.o 1
56.h odd 2 1 4704.2.a.bf 1
56.j odd 6 2 672.2.q.a 2
56.m even 6 2 672.2.q.f yes 2
168.ba even 6 2 2016.2.s.n 2
168.be odd 6 2 2016.2.s.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.a 2 56.j odd 6 2
672.2.q.f yes 2 56.m even 6 2
1344.2.q.k 2 28.f even 6 2
1344.2.q.u 2 7.d odd 6 2
2016.2.s.k 2 168.be odd 6 2
2016.2.s.n 2 168.ba even 6 2
4704.2.a.b 1 8.b even 2 1
4704.2.a.o 1 56.e even 2 1
4704.2.a.s 1 8.d odd 2 1
4704.2.a.bf 1 56.h odd 2 1
9408.2.a.e 1 7.b odd 2 1
9408.2.a.bl 1 4.b odd 2 1
9408.2.a.bt 1 28.d even 2 1
9408.2.a.dc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} - 3 \)
\( T_{11} - 1 \)
\( T_{13} + 4 \)
\( T_{17} + 4 \)
\( T_{19} \)
\( T_{31} + 11 \)

Hecke characteristic polynomials

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