Properties

Label 7200.2.a.bg
Level $7200$
Weight $2$
Character orbit 7200.a
Self dual yes
Analytic conductor $57.492$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.4922894553\)
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

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q + 4q^{11} - 2q^{13} - 2q^{17} - 8q^{19} - 4q^{23} + 6q^{29} - 2q^{37} + 6q^{41} + 4q^{43} + 12q^{47} - 7q^{49} - 6q^{53} + 12q^{59} + 14q^{61} - 12q^{67} - 2q^{73} + 8q^{79} + 4q^{83} - 2q^{89} + 14q^{97} + O(q^{100}) \)

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 0 0 0 0 0 0 0 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 7200.2.a.bg 1
3.b odd 2 1 2400.2.a.y 1
4.b odd 2 1 7200.2.a.u 1
5.b even 2 1 1440.2.a.k 1
5.c odd 4 2 7200.2.f.bb 2
12.b even 2 1 2400.2.a.j 1
15.d odd 2 1 480.2.a.b 1
15.e even 4 2 2400.2.f.e 2
20.d odd 2 1 1440.2.a.j 1
20.e even 4 2 7200.2.f.b 2
24.f even 2 1 4800.2.a.ca 1
24.h odd 2 1 4800.2.a.u 1
40.e odd 2 1 2880.2.a.j 1
40.f even 2 1 2880.2.a.i 1
60.h even 2 1 480.2.a.e yes 1
60.l odd 4 2 2400.2.f.n 2
120.i odd 2 1 960.2.a.o 1
120.m even 2 1 960.2.a.f 1
120.q odd 4 2 4800.2.f.j 2
120.w even 4 2 4800.2.f.ba 2
240.t even 4 2 3840.2.k.k 2
240.bm odd 4 2 3840.2.k.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.b 1 15.d odd 2 1
480.2.a.e yes 1 60.h even 2 1
960.2.a.f 1 120.m even 2 1
960.2.a.o 1 120.i odd 2 1
1440.2.a.j 1 20.d odd 2 1
1440.2.a.k 1 5.b even 2 1
2400.2.a.j 1 12.b even 2 1
2400.2.a.y 1 3.b odd 2 1
2400.2.f.e 2 15.e even 4 2
2400.2.f.n 2 60.l odd 4 2
2880.2.a.i 1 40.f even 2 1
2880.2.a.j 1 40.e odd 2 1
3840.2.k.k 2 240.t even 4 2
3840.2.k.p 2 240.bm odd 4 2
4800.2.a.u 1 24.h odd 2 1
4800.2.a.ca 1 24.f even 2 1
4800.2.f.j 2 120.q odd 4 2
4800.2.f.ba 2 120.w even 4 2
7200.2.a.u 1 4.b odd 2 1
7200.2.a.bg 1 1.a even 1 1 trivial
7200.2.f.b 2 20.e even 4 2
7200.2.f.bb 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7200))\):

\( T_{7} \)
\( T_{11} - 4 \)
\( T_{13} + 2 \)
\( T_{17} + 2 \)
\( T_{19} + 8 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

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