Properties

Label 9408.2.a.i
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 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{5} + q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{5} + q^{9} - 2q^{11} - 3q^{13} + 2q^{15} - 8q^{17} - q^{19} + 8q^{23} - q^{25} - q^{27} - 4q^{29} - 3q^{31} + 2q^{33} + q^{37} + 3q^{39} - 6q^{41} - 11q^{43} - 2q^{45} - 6q^{47} + 8q^{51} + 12q^{53} + 4q^{55} + q^{57} + 4q^{59} - 6q^{61} + 6q^{65} - 13q^{67} - 8q^{69} - 10q^{71} + 11q^{73} + q^{75} - 3q^{79} + q^{81} + 2q^{83} + 16q^{85} + 4q^{87} + 3q^{93} + 2q^{95} - 10q^{97} - 2q^{99} + O(q^{100}) \)

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 −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.i 1
4.b odd 2 1 9408.2.a.bx 1
7.b odd 2 1 9408.2.a.cx 1
7.d odd 6 2 1344.2.q.b 2
8.b even 2 1 588.2.a.f 1
8.d odd 2 1 2352.2.a.k 1
24.f even 2 1 7056.2.a.o 1
24.h odd 2 1 1764.2.a.c 1
28.d even 2 1 9408.2.a.bi 1
28.f even 6 2 1344.2.q.n 2
56.e even 2 1 2352.2.a.o 1
56.h odd 2 1 588.2.a.a 1
56.j odd 6 2 84.2.i.a 2
56.k odd 6 2 2352.2.q.q 2
56.m even 6 2 336.2.q.c 2
56.p even 6 2 588.2.i.b 2
168.e odd 2 1 7056.2.a.bs 1
168.i even 2 1 1764.2.a.h 1
168.s odd 6 2 1764.2.k.j 2
168.ba even 6 2 252.2.k.a 2
168.be odd 6 2 1008.2.s.c 2
280.bk odd 6 2 2100.2.q.b 2
280.bv even 12 4 2100.2.bc.a 4
504.y even 6 2 2268.2.l.g 2
504.bp odd 6 2 2268.2.i.g 2
504.ca even 6 2 2268.2.i.b 2
504.cw odd 6 2 2268.2.l.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 56.j odd 6 2
252.2.k.a 2 168.ba even 6 2
336.2.q.c 2 56.m even 6 2
588.2.a.a 1 56.h odd 2 1
588.2.a.f 1 8.b even 2 1
588.2.i.b 2 56.p even 6 2
1008.2.s.c 2 168.be odd 6 2
1344.2.q.b 2 7.d odd 6 2
1344.2.q.n 2 28.f even 6 2
1764.2.a.c 1 24.h odd 2 1
1764.2.a.h 1 168.i even 2 1
1764.2.k.j 2 168.s odd 6 2
2100.2.q.b 2 280.bk odd 6 2
2100.2.bc.a 4 280.bv even 12 4
2268.2.i.b 2 504.ca even 6 2
2268.2.i.g 2 504.bp odd 6 2
2268.2.l.b 2 504.cw odd 6 2
2268.2.l.g 2 504.y even 6 2
2352.2.a.k 1 8.d odd 2 1
2352.2.a.o 1 56.e even 2 1
2352.2.q.q 2 56.k odd 6 2
7056.2.a.o 1 24.f even 2 1
7056.2.a.bs 1 168.e odd 2 1
9408.2.a.i 1 1.a even 1 1 trivial
9408.2.a.bi 1 28.d even 2 1
9408.2.a.bx 1 4.b odd 2 1
9408.2.a.cx 1 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} + 2 \)
\( T_{11} + 2 \)
\( T_{13} + 3 \)
\( T_{17} + 8 \)
\( T_{19} + 1 \)
\( T_{31} + 3 \)