Properties

Label 7200.2.a.ba
Level $7200$
Weight $2$
Character orbit 7200.a
Self dual yes
Analytic conductor $57.492$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{13} + 6 q^{17} + 4 q^{19} - 8 q^{23} + 2 q^{29} - 4 q^{31} - 10 q^{37} - 2 q^{41} - 4 q^{43} - 8 q^{47} - 7 q^{49} - 2 q^{53} + 8 q^{59} - 2 q^{61} - 12 q^{67} + 8 q^{71} + 14 q^{73} + 12 q^{79} + 4 q^{83} + 14 q^{89} - 2 q^{97}+O(q^{100}) \) 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 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.ba 1
3.b odd 2 1 2400.2.a.i 1
4.b odd 2 1 7200.2.a.z 1
5.b even 2 1 1440.2.a.d 1
5.c odd 4 2 7200.2.f.k 2
12.b even 2 1 2400.2.a.z 1
15.d odd 2 1 480.2.a.g yes 1
15.e even 4 2 2400.2.f.g 2
20.d odd 2 1 1440.2.a.c 1
20.e even 4 2 7200.2.f.s 2
24.f even 2 1 4800.2.a.s 1
24.h odd 2 1 4800.2.a.cb 1
40.e odd 2 1 2880.2.a.ba 1
40.f even 2 1 2880.2.a.z 1
60.h even 2 1 480.2.a.d 1
60.l odd 4 2 2400.2.f.l 2
120.i odd 2 1 960.2.a.b 1
120.m even 2 1 960.2.a.k 1
120.q odd 4 2 4800.2.f.o 2
120.w even 4 2 4800.2.f.v 2
240.t even 4 2 3840.2.k.n 2
240.bm odd 4 2 3840.2.k.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.d 1 60.h even 2 1
480.2.a.g yes 1 15.d odd 2 1
960.2.a.b 1 120.i odd 2 1
960.2.a.k 1 120.m even 2 1
1440.2.a.c 1 20.d odd 2 1
1440.2.a.d 1 5.b even 2 1
2400.2.a.i 1 3.b odd 2 1
2400.2.a.z 1 12.b even 2 1
2400.2.f.g 2 15.e even 4 2
2400.2.f.l 2 60.l odd 4 2
2880.2.a.z 1 40.f even 2 1
2880.2.a.ba 1 40.e odd 2 1
3840.2.k.n 2 240.t even 4 2
3840.2.k.s 2 240.bm odd 4 2
4800.2.a.s 1 24.f even 2 1
4800.2.a.cb 1 24.h odd 2 1
4800.2.f.o 2 120.q odd 4 2
4800.2.f.v 2 120.w even 4 2
7200.2.a.z 1 4.b odd 2 1
7200.2.a.ba 1 1.a even 1 1 trivial
7200.2.f.k 2 5.c odd 4 2
7200.2.f.s 2 20.e even 4 2

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T - 12 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 14 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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