Properties

Label 9408.2.a.cq
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 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} + q^{9} + 3q^{11} - 4q^{13} + q^{15} - 4q^{19} - 8q^{23} - 4q^{25} + q^{27} + 3q^{29} + 5q^{31} + 3q^{33} - 8q^{37} - 4q^{39} + 8q^{41} + 6q^{43} + q^{45} - 10q^{47} - 9q^{53} + 3q^{55} - 4q^{57} - 5q^{59} + 10q^{61} - 4q^{65} + 6q^{67} - 8q^{69} - 10q^{71} + 2q^{73} - 4q^{75} - 11q^{79} + q^{81} + 7q^{83} + 3q^{87} - 18q^{89} + 5q^{93} - 4q^{95} - 17q^{97} + 3q^{99} + O(q^{100}) \)

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:

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.cq 1
4.b odd 2 1 9408.2.a.ba 1
7.b odd 2 1 9408.2.a.p 1
7.c even 3 2 1344.2.q.d 2
8.b even 2 1 2352.2.a.g 1
8.d odd 2 1 1176.2.a.g 1
24.f even 2 1 3528.2.a.q 1
24.h odd 2 1 7056.2.a.bk 1
28.d even 2 1 9408.2.a.cf 1
28.g odd 6 2 1344.2.q.o 2
56.e even 2 1 1176.2.a.c 1
56.h odd 2 1 2352.2.a.u 1
56.j odd 6 2 2352.2.q.f 2
56.k odd 6 2 168.2.q.a 2
56.m even 6 2 1176.2.q.g 2
56.p even 6 2 336.2.q.e 2
168.e odd 2 1 3528.2.a.i 1
168.i even 2 1 7056.2.a.t 1
168.s odd 6 2 1008.2.s.f 2
168.v even 6 2 504.2.s.d 2
168.be odd 6 2 3528.2.s.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.a 2 56.k odd 6 2
336.2.q.e 2 56.p even 6 2
504.2.s.d 2 168.v even 6 2
1008.2.s.f 2 168.s odd 6 2
1176.2.a.c 1 56.e even 2 1
1176.2.a.g 1 8.d odd 2 1
1176.2.q.g 2 56.m even 6 2
1344.2.q.d 2 7.c even 3 2
1344.2.q.o 2 28.g odd 6 2
2352.2.a.g 1 8.b even 2 1
2352.2.a.u 1 56.h odd 2 1
2352.2.q.f 2 56.j odd 6 2
3528.2.a.i 1 168.e odd 2 1
3528.2.a.q 1 24.f even 2 1
3528.2.s.p 2 168.be odd 6 2
7056.2.a.t 1 168.i even 2 1
7056.2.a.bk 1 24.h odd 2 1
9408.2.a.p 1 7.b odd 2 1
9408.2.a.ba 1 4.b odd 2 1
9408.2.a.cf 1 28.d even 2 1
9408.2.a.cq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ \( 1 - T + 5 T^{2} \)
$7$ 1
$11$ \( 1 - 3 T + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 8 T + 23 T^{2} \)
$29$ \( 1 - 3 T + 29 T^{2} \)
$31$ \( 1 - 5 T + 31 T^{2} \)
$37$ \( 1 + 8 T + 37 T^{2} \)
$41$ \( 1 - 8 T + 41 T^{2} \)
$43$ \( 1 - 6 T + 43 T^{2} \)
$47$ \( 1 + 10 T + 47 T^{2} \)
$53$ \( 1 + 9 T + 53 T^{2} \)
$59$ \( 1 + 5 T + 59 T^{2} \)
$61$ \( 1 - 10 T + 61 T^{2} \)
$67$ \( 1 - 6 T + 67 T^{2} \)
$71$ \( 1 + 10 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 + 11 T + 79 T^{2} \)
$83$ \( 1 - 7 T + 83 T^{2} \)
$89$ \( 1 + 18 T + 89 T^{2} \)
$97$ \( 1 + 17 T + 97 T^{2} \)
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