Properties

Label 7200.2.a.v
Level $7200$
Weight $2$
Character orbit 7200.a
Self dual yes
Analytic conductor $57.492$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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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 32)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + O(q^{10}) \) \( q - 6q^{13} + 2q^{17} + 10q^{29} + 2q^{37} - 10q^{41} - 7q^{49} + 14q^{53} - 10q^{61} + 6q^{73} - 10q^{89} - 18q^{97} + O(q^{100}) \)

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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.a.v 1
3.b odd 2 1 800.2.a.d 1
4.b odd 2 1 CM 7200.2.a.v 1
5.b even 2 1 288.2.a.d 1
5.c odd 4 2 7200.2.f.m 2
12.b even 2 1 800.2.a.d 1
15.d odd 2 1 32.2.a.a 1
15.e even 4 2 800.2.c.e 2
20.d odd 2 1 288.2.a.d 1
20.e even 4 2 7200.2.f.m 2
24.f even 2 1 1600.2.a.n 1
24.h odd 2 1 1600.2.a.n 1
40.e odd 2 1 576.2.a.c 1
40.f even 2 1 576.2.a.c 1
45.h odd 6 2 2592.2.i.t 2
45.j even 6 2 2592.2.i.e 2
60.h even 2 1 32.2.a.a 1
60.l odd 4 2 800.2.c.e 2
80.k odd 4 2 2304.2.d.j 2
80.q even 4 2 2304.2.d.j 2
105.g even 2 1 1568.2.a.e 1
105.o odd 6 2 1568.2.i.g 2
105.p even 6 2 1568.2.i.f 2
120.i odd 2 1 64.2.a.a 1
120.m even 2 1 64.2.a.a 1
120.q odd 4 2 1600.2.c.l 2
120.w even 4 2 1600.2.c.l 2
165.d even 2 1 3872.2.a.f 1
180.n even 6 2 2592.2.i.t 2
180.p odd 6 2 2592.2.i.e 2
195.e odd 2 1 5408.2.a.g 1
240.t even 4 2 256.2.b.b 2
240.bm odd 4 2 256.2.b.b 2
255.h odd 2 1 9248.2.a.f 1
420.o odd 2 1 1568.2.a.e 1
420.ba even 6 2 1568.2.i.g 2
420.be odd 6 2 1568.2.i.f 2
480.bs even 8 4 1024.2.e.j 4
480.bu odd 8 4 1024.2.e.j 4
660.g odd 2 1 3872.2.a.f 1
780.d even 2 1 5408.2.a.g 1
840.b odd 2 1 3136.2.a.m 1
840.u even 2 1 3136.2.a.m 1
1020.b even 2 1 9248.2.a.f 1
1320.b odd 2 1 7744.2.a.v 1
1320.u even 2 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 15.d odd 2 1
32.2.a.a 1 60.h even 2 1
64.2.a.a 1 120.i odd 2 1
64.2.a.a 1 120.m even 2 1
256.2.b.b 2 240.t even 4 2
256.2.b.b 2 240.bm odd 4 2
288.2.a.d 1 5.b even 2 1
288.2.a.d 1 20.d odd 2 1
576.2.a.c 1 40.e odd 2 1
576.2.a.c 1 40.f even 2 1
800.2.a.d 1 3.b odd 2 1
800.2.a.d 1 12.b even 2 1
800.2.c.e 2 15.e even 4 2
800.2.c.e 2 60.l odd 4 2
1024.2.e.j 4 480.bs even 8 4
1024.2.e.j 4 480.bu odd 8 4
1568.2.a.e 1 105.g even 2 1
1568.2.a.e 1 420.o odd 2 1
1568.2.i.f 2 105.p even 6 2
1568.2.i.f 2 420.be odd 6 2
1568.2.i.g 2 105.o odd 6 2
1568.2.i.g 2 420.ba even 6 2
1600.2.a.n 1 24.f even 2 1
1600.2.a.n 1 24.h odd 2 1
1600.2.c.l 2 120.q odd 4 2
1600.2.c.l 2 120.w even 4 2
2304.2.d.j 2 80.k odd 4 2
2304.2.d.j 2 80.q even 4 2
2592.2.i.e 2 45.j even 6 2
2592.2.i.e 2 180.p odd 6 2
2592.2.i.t 2 45.h odd 6 2
2592.2.i.t 2 180.n even 6 2
3136.2.a.m 1 840.b odd 2 1
3136.2.a.m 1 840.u even 2 1
3872.2.a.f 1 165.d even 2 1
3872.2.a.f 1 660.g odd 2 1
5408.2.a.g 1 195.e odd 2 1
5408.2.a.g 1 780.d even 2 1
7200.2.a.v 1 1.a even 1 1 trivial
7200.2.a.v 1 4.b odd 2 1 CM
7200.2.f.m 2 5.c odd 4 2
7200.2.f.m 2 20.e even 4 2
7744.2.a.v 1 1320.b odd 2 1
7744.2.a.v 1 1320.u even 2 1
9248.2.a.f 1 255.h odd 2 1
9248.2.a.f 1 1020.b even 2 1

Hecke kernels

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

\( T_{7} \)
\( T_{11} \)
\( T_{13} + 6 \)
\( T_{17} - 2 \)
\( T_{19} \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 6 + T \)
$17$ \( -2 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -10 + T \)
$31$ \( T \)
$37$ \( -2 + T \)
$41$ \( 10 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -14 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( 10 + T \)
$97$ \( 18 + T \)
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