Properties

Label 9408.2.a.o
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

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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{9} + O(q^{10}) \) \( q - q^{3} - q^{5} + q^{9} + q^{11} + q^{15} - 8q^{17} - 4q^{19} - 4q^{23} - 4q^{25} - q^{27} + 5q^{29} - 7q^{31} - q^{33} - 8q^{37} + 4q^{41} + 10q^{43} - q^{45} - 6q^{47} + 8q^{51} + q^{53} - q^{55} + 4q^{57} + 9q^{59} + 2q^{61} + 2q^{67} + 4q^{69} - 6q^{71} + 2q^{73} + 4q^{75} + 9q^{79} + q^{81} - 3q^{83} + 8q^{85} - 5q^{87} - 6q^{89} + 7q^{93} + 4q^{95} - q^{97} + q^{99} + O(q^{100}) \)

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 −1.00000 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.o 1
4.b odd 2 1 9408.2.a.cg 1
7.b odd 2 1 9408.2.a.cp 1
7.c even 3 2 1344.2.q.r 2
8.b even 2 1 4704.2.a.bc 1
8.d odd 2 1 4704.2.a.l 1
28.d even 2 1 9408.2.a.bb 1
28.g odd 6 2 1344.2.q.h 2
56.e even 2 1 4704.2.a.w 1
56.h odd 2 1 4704.2.a.f 1
56.k odd 6 2 672.2.q.g yes 2
56.p even 6 2 672.2.q.b 2
168.s odd 6 2 2016.2.s.i 2
168.v even 6 2 2016.2.s.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.b 2 56.p even 6 2
672.2.q.g yes 2 56.k odd 6 2
1344.2.q.h 2 28.g odd 6 2
1344.2.q.r 2 7.c even 3 2
2016.2.s.i 2 168.s odd 6 2
2016.2.s.j 2 168.v even 6 2
4704.2.a.f 1 56.h odd 2 1
4704.2.a.l 1 8.d odd 2 1
4704.2.a.w 1 56.e even 2 1
4704.2.a.bc 1 8.b even 2 1
9408.2.a.o 1 1.a even 1 1 trivial
9408.2.a.bb 1 28.d even 2 1
9408.2.a.cg 1 4.b odd 2 1
9408.2.a.cp 1 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} + 1 \)
\( T_{11} - 1 \)
\( T_{13} \)
\( T_{17} + 8 \)
\( T_{19} + 4 \)
\( T_{31} + 7 \)

Hecke characteristic polynomials

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