Properties

Label 7200.2.a.cr
Level $7200$
Weight $2$
Character orbit 7200.a
Self dual yes
Analytic conductor $57.492$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $2$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 + \beta ) q^{7} +O(q^{10})\) \( q + ( 3 + \beta ) q^{7} + ( -1 - 3 \beta ) q^{23} + 6 q^{29} -2 \beta q^{41} + ( 9 - \beta ) q^{43} + ( -7 + 3 \beta ) q^{47} + ( 7 + 6 \beta ) q^{49} + 6 \beta q^{61} + ( 3 + 5 \beta ) q^{67} + ( 11 - 3 \beta ) q^{83} -6 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{7} + O(q^{10}) \) \( 2 q + 6 q^{7} - 2 q^{23} + 12 q^{29} + 18 q^{43} - 14 q^{47} + 14 q^{49} + 6 q^{67} + 22 q^{83} - 12 q^{89} + O(q^{100}) \)

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
−0.618034
1.61803
0 0 0 0 0 0.763932 0 0 0
1.2 0 0 0 0 0 5.23607 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.a.cr 2
3.b odd 2 1 800.2.a.n 2
4.b odd 2 1 7200.2.a.cb 2
5.b even 2 1 7200.2.a.cb 2
5.c odd 4 2 1440.2.f.i 4
12.b even 2 1 800.2.a.j 2
15.d odd 2 1 800.2.a.j 2
15.e even 4 2 160.2.c.b 4
20.d odd 2 1 CM 7200.2.a.cr 2
20.e even 4 2 1440.2.f.i 4
24.f even 2 1 1600.2.a.bd 2
24.h odd 2 1 1600.2.a.z 2
40.i odd 4 2 2880.2.f.w 4
40.k even 4 2 2880.2.f.w 4
60.h even 2 1 800.2.a.n 2
60.l odd 4 2 160.2.c.b 4
120.i odd 2 1 1600.2.a.bd 2
120.m even 2 1 1600.2.a.z 2
120.q odd 4 2 320.2.c.d 4
120.w even 4 2 320.2.c.d 4
240.z odd 4 2 1280.2.f.h 4
240.bb even 4 2 1280.2.f.g 4
240.bd odd 4 2 1280.2.f.g 4
240.bf even 4 2 1280.2.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 15.e even 4 2
160.2.c.b 4 60.l odd 4 2
320.2.c.d 4 120.q odd 4 2
320.2.c.d 4 120.w even 4 2
800.2.a.j 2 12.b even 2 1
800.2.a.j 2 15.d odd 2 1
800.2.a.n 2 3.b odd 2 1
800.2.a.n 2 60.h even 2 1
1280.2.f.g 4 240.bb even 4 2
1280.2.f.g 4 240.bd odd 4 2
1280.2.f.h 4 240.z odd 4 2
1280.2.f.h 4 240.bf even 4 2
1440.2.f.i 4 5.c odd 4 2
1440.2.f.i 4 20.e even 4 2
1600.2.a.z 2 24.h odd 2 1
1600.2.a.z 2 120.m even 2 1
1600.2.a.bd 2 24.f even 2 1
1600.2.a.bd 2 120.i odd 2 1
2880.2.f.w 4 40.i odd 4 2
2880.2.f.w 4 40.k even 4 2
7200.2.a.cb 2 4.b odd 2 1
7200.2.a.cb 2 5.b even 2 1
7200.2.a.cr 2 1.a even 1 1 trivial
7200.2.a.cr 2 20.d odd 2 1 CM

Hecke kernels

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

\( T_{7}^{2} - 6 T_{7} + 4 \)
\( T_{11} \)
\( T_{13} \)
\( T_{17} \)
\( T_{19} \)
\( T_{23}^{2} + 2 T_{23} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 - 6 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( -44 + 2 T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( -20 + T^{2} \)
$43$ \( 76 - 18 T + T^{2} \)
$47$ \( 4 + 14 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( -180 + T^{2} \)
$67$ \( -116 - 6 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( 76 - 22 T + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( T^{2} \)
show more
show less