Properties

Label 4800.2.a.bn
Level $4800$
Weight $2$
Character orbit 4800.a
Self dual yes
Analytic conductor $38.328$
Analytic rank $1$
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: \(1\)
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

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

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.bn 1
4.b odd 2 1 4800.2.a.bf 1
5.b even 2 1 4800.2.a.bj 1
5.c odd 4 2 960.2.f.f 2
8.b even 2 1 300.2.a.a 1
8.d odd 2 1 1200.2.a.s 1
15.e even 4 2 2880.2.f.l 2
20.d odd 2 1 4800.2.a.bk 1
20.e even 4 2 960.2.f.c 2
24.f even 2 1 3600.2.a.bm 1
24.h odd 2 1 900.2.a.a 1
40.e odd 2 1 1200.2.a.a 1
40.f even 2 1 300.2.a.d 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.o 2
80.j even 4 2 3840.2.d.b 2
80.s even 4 2 3840.2.d.be 2
80.t odd 4 2 3840.2.d.r 2
120.i odd 2 1 900.2.a.h 1
120.m even 2 1 3600.2.a.d 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 8.b even 2 1
300.2.a.d 1 40.f even 2 1
720.2.f.c 2 120.q odd 4 2
900.2.a.a 1 24.h odd 2 1
900.2.a.h 1 120.i 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 40.e odd 2 1
1200.2.a.s 1 8.d 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 1 120.m even 2 1
3600.2.a.bm 1 24.f even 2 1
3840.2.d.b 2 80.j even 4 2
3840.2.d.o 2 80.i odd 4 2
3840.2.d.r 2 80.t odd 4 2
3840.2.d.be 2 80.s even 4 2
4800.2.a.bf 1 4.b odd 2 1
4800.2.a.bj 1 5.b even 2 1
4800.2.a.bk 1 20.d odd 2 1
4800.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4800))\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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