Properties

Label 4800.2.a.bj
Level $4800$
Weight $2$
Character orbit 4800.a
Self dual yes
Analytic conductor $38.328$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 4q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + 4q^{7} + q^{9} + 4q^{11} + 4q^{17} - 4q^{21} + 4q^{23} - q^{27} + 6q^{29} + 4q^{31} - 4q^{33} + 8q^{37} - 10q^{41} - 4q^{43} - 4q^{47} + 9q^{49} - 4q^{51} + 12q^{53} - 4q^{59} - 2q^{61} + 4q^{63} + 4q^{67} - 4q^{69} - 8q^{73} + 16q^{77} - 12q^{79} + q^{81} - 4q^{83} - 6q^{87} - 10q^{89} - 4q^{93} + 8q^{97} + 4q^{99} + O(q^{100}) \)

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 0 0 4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)

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 4800.2.a.bj 1
4.b odd 2 1 4800.2.a.bk 1
5.b even 2 1 4800.2.a.bn 1
5.c odd 4 2 960.2.f.f 2
8.b even 2 1 300.2.a.d 1
8.d odd 2 1 1200.2.a.a 1
15.e even 4 2 2880.2.f.l 2
20.d odd 2 1 4800.2.a.bf 1
20.e even 4 2 960.2.f.c 2
24.f even 2 1 3600.2.a.d 1
24.h odd 2 1 900.2.a.h 1
40.e odd 2 1 1200.2.a.s 1
40.f even 2 1 300.2.a.a 1
40.i odd 4 2 60.2.d.a 2
40.k even 4 2 240.2.f.b 2
60.l odd 4 2 2880.2.f.p 2
80.i odd 4 2 3840.2.d.r 2
80.j even 4 2 3840.2.d.be 2
80.s even 4 2 3840.2.d.b 2
80.t odd 4 2 3840.2.d.o 2
120.i odd 2 1 900.2.a.a 1
120.m even 2 1 3600.2.a.bm 1
120.q odd 4 2 720.2.f.c 2
120.w even 4 2 180.2.d.a 2
280.s even 4 2 2940.2.k.c 2
280.bt odd 12 4 2940.2.bb.d 4
280.bv even 12 4 2940.2.bb.e 4
360.br even 12 4 1620.2.r.d 4
360.bu odd 12 4 1620.2.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 40.i odd 4 2
180.2.d.a 2 120.w even 4 2
240.2.f.b 2 40.k even 4 2
300.2.a.a 1 40.f even 2 1
300.2.a.d 1 8.b even 2 1
720.2.f.c 2 120.q odd 4 2
900.2.a.a 1 120.i odd 2 1
900.2.a.h 1 24.h odd 2 1
960.2.f.c 2 20.e even 4 2
960.2.f.f 2 5.c odd 4 2
1200.2.a.a 1 8.d odd 2 1
1200.2.a.s 1 40.e odd 2 1
1620.2.r.c 4 360.bu odd 12 4
1620.2.r.d 4 360.br even 12 4
2880.2.f.l 2 15.e even 4 2
2880.2.f.p 2 60.l odd 4 2
2940.2.k.c 2 280.s even 4 2
2940.2.bb.d 4 280.bt odd 12 4
2940.2.bb.e 4 280.bv even 12 4
3600.2.a.d 1 24.f even 2 1
3600.2.a.bm 1 120.m even 2 1
3840.2.d.b 2 80.s even 4 2
3840.2.d.o 2 80.t odd 4 2
3840.2.d.r 2 80.i odd 4 2
3840.2.d.be 2 80.j even 4 2
4800.2.a.bf 1 20.d odd 2 1
4800.2.a.bj 1 1.a even 1 1 trivial
4800.2.a.bk 1 4.b odd 2 1
4800.2.a.bn 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4800))\):

\( T_{7} - 4 \)
\( T_{11} - 4 \)
\( T_{13} \)
\( T_{19} \)
\( T_{23} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( -4 + T \)
$11$ \( -4 + T \)
$13$ \( T \)
$17$ \( -4 + T \)
$19$ \( T \)
$23$ \( -4 + T \)
$29$ \( -6 + T \)
$31$ \( -4 + T \)
$37$ \( -8 + T \)
$41$ \( 10 + T \)
$43$ \( 4 + T \)
$47$ \( 4 + T \)
$53$ \( -12 + T \)
$59$ \( 4 + T \)
$61$ \( 2 + T \)
$67$ \( -4 + T \)
$71$ \( T \)
$73$ \( 8 + T \)
$79$ \( 12 + T \)
$83$ \( 4 + T \)
$89$ \( 10 + T \)
$97$ \( -8 + T \)
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