Properties

Label 4800.2.a.l
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 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{7} + q^{9} - 2q^{11} + 6q^{13} - 2q^{17} + 2q^{21} + 4q^{23} - q^{27} - 8q^{31} + 2q^{33} + 2q^{37} - 6q^{39} + 2q^{41} - 4q^{43} + 8q^{47} - 3q^{49} + 2q^{51} + 6q^{53} - 10q^{59} - 2q^{61} - 2q^{63} - 8q^{67} - 4q^{69} + 12q^{71} + 4q^{73} + 4q^{77} + q^{81} - 4q^{83} - 10q^{89} - 12q^{91} + 8q^{93} + 8q^{97} - 2q^{99} + O(q^{100}) \)

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 0 0 −2.00000 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.l 1
4.b odd 2 1 4800.2.a.cj 1
5.b even 2 1 4800.2.a.cg 1
5.c odd 4 2 960.2.f.h 2
8.b even 2 1 150.2.a.c 1
8.d odd 2 1 1200.2.a.g 1
15.e even 4 2 2880.2.f.e 2
20.d odd 2 1 4800.2.a.m 1
20.e even 4 2 960.2.f.i 2
24.f even 2 1 3600.2.a.bg 1
24.h odd 2 1 450.2.a.b 1
40.e odd 2 1 1200.2.a.m 1
40.f even 2 1 150.2.a.a 1
40.i odd 4 2 30.2.c.a 2
40.k even 4 2 240.2.f.a 2
56.h odd 2 1 7350.2.a.cc 1
60.l odd 4 2 2880.2.f.c 2
80.i odd 4 2 3840.2.d.y 2
80.j even 4 2 3840.2.d.x 2
80.s even 4 2 3840.2.d.j 2
80.t odd 4 2 3840.2.d.g 2
120.i odd 2 1 450.2.a.f 1
120.m even 2 1 3600.2.a.o 1
120.q odd 4 2 720.2.f.f 2
120.w even 4 2 90.2.c.a 2
280.c odd 2 1 7350.2.a.bg 1
280.s even 4 2 1470.2.g.g 2
280.bt odd 12 4 1470.2.n.h 4
280.bv even 12 4 1470.2.n.a 4
360.br even 12 4 810.2.i.b 4
360.bu odd 12 4 810.2.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 40.i odd 4 2
90.2.c.a 2 120.w even 4 2
150.2.a.a 1 40.f even 2 1
150.2.a.c 1 8.b even 2 1
240.2.f.a 2 40.k even 4 2
450.2.a.b 1 24.h odd 2 1
450.2.a.f 1 120.i odd 2 1
720.2.f.f 2 120.q odd 4 2
810.2.i.b 4 360.br even 12 4
810.2.i.e 4 360.bu odd 12 4
960.2.f.h 2 5.c odd 4 2
960.2.f.i 2 20.e even 4 2
1200.2.a.g 1 8.d odd 2 1
1200.2.a.m 1 40.e odd 2 1
1470.2.g.g 2 280.s even 4 2
1470.2.n.a 4 280.bv even 12 4
1470.2.n.h 4 280.bt odd 12 4
2880.2.f.c 2 60.l odd 4 2
2880.2.f.e 2 15.e even 4 2
3600.2.a.o 1 120.m even 2 1
3600.2.a.bg 1 24.f even 2 1
3840.2.d.g 2 80.t odd 4 2
3840.2.d.j 2 80.s even 4 2
3840.2.d.x 2 80.j even 4 2
3840.2.d.y 2 80.i odd 4 2
4800.2.a.l 1 1.a even 1 1 trivial
4800.2.a.m 1 20.d odd 2 1
4800.2.a.cg 1 5.b even 2 1
4800.2.a.cj 1 4.b odd 2 1
7350.2.a.bg 1 280.c odd 2 1
7350.2.a.cc 1 56.h odd 2 1

Hecke kernels

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

\( T_{7} + 2 \)
\( T_{11} + 2 \)
\( T_{13} - 6 \)
\( T_{19} \)
\( T_{23} - 4 \)