Properties

Label 9408.2.a.cr
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 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.cr 1
4.b odd 2 1 9408.2.a.z 1
7.b odd 2 1 9408.2.a.q 1
7.d odd 6 2 1344.2.q.s 2
8.b even 2 1 2352.2.a.f 1
8.d odd 2 1 294.2.a.f 1
24.f even 2 1 882.2.a.d 1
24.h odd 2 1 7056.2.a.bl 1
28.d even 2 1 9408.2.a.ce 1
28.f even 6 2 1344.2.q.g 2
40.e odd 2 1 7350.2.a.q 1
56.e even 2 1 294.2.a.e 1
56.h odd 2 1 2352.2.a.t 1
56.j odd 6 2 336.2.q.b 2
56.k odd 6 2 294.2.e.b 2
56.m even 6 2 42.2.e.a 2
56.p even 6 2 2352.2.q.u 2
168.e odd 2 1 882.2.a.c 1
168.i even 2 1 7056.2.a.w 1
168.v even 6 2 882.2.g.i 2
168.ba even 6 2 1008.2.s.k 2
168.be odd 6 2 126.2.g.c 2
280.n even 2 1 7350.2.a.bl 1
280.ba even 6 2 1050.2.i.l 2
280.bp odd 12 4 1050.2.o.a 4
504.u odd 6 2 1134.2.h.l 2
504.bf even 6 2 1134.2.e.l 2
504.cm odd 6 2 1134.2.e.e 2
504.cz even 6 2 1134.2.h.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 56.m even 6 2
126.2.g.c 2 168.be odd 6 2
294.2.a.e 1 56.e even 2 1
294.2.a.f 1 8.d odd 2 1
294.2.e.b 2 56.k odd 6 2
336.2.q.b 2 56.j odd 6 2
882.2.a.c 1 168.e odd 2 1
882.2.a.d 1 24.f even 2 1
882.2.g.i 2 168.v even 6 2
1008.2.s.k 2 168.ba even 6 2
1050.2.i.l 2 280.ba even 6 2
1050.2.o.a 4 280.bp odd 12 4
1134.2.e.e 2 504.cm odd 6 2
1134.2.e.l 2 504.bf even 6 2
1134.2.h.e 2 504.cz even 6 2
1134.2.h.l 2 504.u odd 6 2
1344.2.q.g 2 28.f even 6 2
1344.2.q.s 2 7.d odd 6 2
2352.2.a.f 1 8.b even 2 1
2352.2.a.t 1 56.h odd 2 1
2352.2.q.u 2 56.p even 6 2
7056.2.a.w 1 168.i even 2 1
7056.2.a.bl 1 24.h odd 2 1
7350.2.a.q 1 40.e odd 2 1
7350.2.a.bl 1 280.n even 2 1
9408.2.a.q 1 7.b odd 2 1
9408.2.a.z 1 4.b odd 2 1
9408.2.a.ce 1 28.d even 2 1
9408.2.a.cr 1 1.a even 1 1 trivial

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

\( T_{5} - 1 \)
\( T_{11} - 5 \)
\( T_{13} \)
\( T_{17} - 4 \)
\( T_{19} + 8 \)
\( T_{31} - 3 \)

Hecke characteristic polynomials

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