Properties

Label 2400.2.a.bf
Level $2400$
Weight $2$
Character orbit 2400.a
Self dual yes
Analytic conductor $19.164$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{7} + q^{9} + 6q^{11} + 2q^{13} + 6q^{17} - 4q^{19} + 2q^{21} - 8q^{23} + q^{27} + 8q^{31} + 6q^{33} - 2q^{37} + 2q^{39} - 6q^{41} + 4q^{43} - 4q^{47} - 3q^{49} + 6q^{51} - 6q^{53} - 4q^{57} + 6q^{59} - 6q^{61} + 2q^{63} - 8q^{69} + 4q^{71} - 12q^{73} + 12q^{77} + 8q^{79} + q^{81} + 12q^{83} + 14q^{89} + 4q^{91} + 8q^{93} - 8q^{97} + 6q^{99} + O(q^{100}) \)

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 2400.2.a.bf 1
3.b odd 2 1 7200.2.a.bl 1
4.b odd 2 1 2400.2.a.c 1
5.b even 2 1 2400.2.a.d 1
5.c odd 4 2 480.2.f.b yes 2
8.b even 2 1 4800.2.a.z 1
8.d odd 2 1 4800.2.a.bu 1
12.b even 2 1 7200.2.a.p 1
15.d odd 2 1 7200.2.a.j 1
15.e even 4 2 1440.2.f.e 2
20.d odd 2 1 2400.2.a.be 1
20.e even 4 2 480.2.f.a 2
40.e odd 2 1 4800.2.a.ba 1
40.f even 2 1 4800.2.a.bt 1
40.i odd 4 2 960.2.f.g 2
40.k even 4 2 960.2.f.j 2
60.h even 2 1 7200.2.a.br 1
60.l odd 4 2 1440.2.f.g 2
80.i odd 4 2 3840.2.d.e 2
80.j even 4 2 3840.2.d.k 2
80.s even 4 2 3840.2.d.u 2
80.t odd 4 2 3840.2.d.bc 2
120.q odd 4 2 2880.2.f.a 2
120.w even 4 2 2880.2.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 20.e even 4 2
480.2.f.b yes 2 5.c odd 4 2
960.2.f.g 2 40.i odd 4 2
960.2.f.j 2 40.k even 4 2
1440.2.f.e 2 15.e even 4 2
1440.2.f.g 2 60.l odd 4 2
2400.2.a.c 1 4.b odd 2 1
2400.2.a.d 1 5.b even 2 1
2400.2.a.be 1 20.d odd 2 1
2400.2.a.bf 1 1.a even 1 1 trivial
2880.2.f.a 2 120.q odd 4 2
2880.2.f.g 2 120.w even 4 2
3840.2.d.e 2 80.i odd 4 2
3840.2.d.k 2 80.j even 4 2
3840.2.d.u 2 80.s even 4 2
3840.2.d.bc 2 80.t odd 4 2
4800.2.a.z 1 8.b even 2 1
4800.2.a.ba 1 40.e odd 2 1
4800.2.a.bt 1 40.f even 2 1
4800.2.a.bu 1 8.d odd 2 1
7200.2.a.j 1 15.d odd 2 1
7200.2.a.p 1 12.b even 2 1
7200.2.a.bl 1 3.b odd 2 1
7200.2.a.br 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2400))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( -6 + T \)
$13$ \( -2 + T \)
$17$ \( -6 + T \)
$19$ \( 4 + T \)
$23$ \( 8 + T \)
$29$ \( T \)
$31$ \( -8 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( 4 + T \)
$53$ \( 6 + T \)
$59$ \( -6 + T \)
$61$ \( 6 + T \)
$67$ \( T \)
$71$ \( -4 + T \)
$73$ \( 12 + T \)
$79$ \( -8 + T \)
$83$ \( -12 + T \)
$89$ \( -14 + T \)
$97$ \( 8 + T \)
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