Properties

Label 2160.2.a.bb
Level $2160$
Weight $2$
Character orbit 2160.a
Self dual yes
Analytic conductor $17.248$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Defining polynomial: \(x^{2} - x - 18\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} -\beta q^{7} +O(q^{10})\) \( q + q^{5} -\beta q^{7} + ( 1 - \beta ) q^{11} + 3 q^{13} + ( 3 - \beta ) q^{17} + ( -2 + \beta ) q^{19} + ( 3 + \beta ) q^{23} + q^{25} + ( 1 - \beta ) q^{29} + ( -7 + \beta ) q^{31} -\beta q^{35} + ( 4 - \beta ) q^{37} + 2 \beta q^{41} + ( -3 + \beta ) q^{43} + ( 1 + \beta ) q^{47} + ( 11 + \beta ) q^{49} + 2 \beta q^{53} + ( 1 - \beta ) q^{55} + 12 q^{59} + ( -4 + \beta ) q^{61} + 3 q^{65} + ( -10 - \beta ) q^{67} + ( 4 + 2 \beta ) q^{71} + \beta q^{73} + 18 q^{77} + 5 q^{79} + 6 q^{83} + ( 3 - \beta ) q^{85} -8 q^{89} -3 \beta q^{91} + ( -2 + \beta ) q^{95} + ( -4 - 3 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} - q^{7} + q^{11} + 6 q^{13} + 5 q^{17} - 3 q^{19} + 7 q^{23} + 2 q^{25} + q^{29} - 13 q^{31} - q^{35} + 7 q^{37} + 2 q^{41} - 5 q^{43} + 3 q^{47} + 23 q^{49} + 2 q^{53} + q^{55} + 24 q^{59} - 7 q^{61} + 6 q^{65} - 21 q^{67} + 10 q^{71} + q^{73} + 36 q^{77} + 10 q^{79} + 12 q^{83} + 5 q^{85} - 16 q^{89} - 3 q^{91} - 3 q^{95} - 11 q^{97} + O(q^{100}) \)

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
4.77200
−3.77200
0 0 0 1.00000 0 −4.77200 0 0 0
1.2 0 0 0 1.00000 0 3.77200 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 2160.2.a.bb 2
3.b odd 2 1 2160.2.a.z 2
4.b odd 2 1 1080.2.a.n yes 2
8.b even 2 1 8640.2.a.cn 2
8.d odd 2 1 8640.2.a.cq 2
12.b even 2 1 1080.2.a.m 2
20.d odd 2 1 5400.2.a.ca 2
20.e even 4 2 5400.2.f.bd 4
24.f even 2 1 8640.2.a.de 2
24.h odd 2 1 8640.2.a.db 2
36.f odd 6 2 3240.2.q.z 4
36.h even 6 2 3240.2.q.bc 4
60.h even 2 1 5400.2.a.cb 2
60.l odd 4 2 5400.2.f.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.m 2 12.b even 2 1
1080.2.a.n yes 2 4.b odd 2 1
2160.2.a.z 2 3.b odd 2 1
2160.2.a.bb 2 1.a even 1 1 trivial
3240.2.q.z 4 36.f odd 6 2
3240.2.q.bc 4 36.h even 6 2
5400.2.a.ca 2 20.d odd 2 1
5400.2.a.cb 2 60.h even 2 1
5400.2.f.bd 4 20.e even 4 2
5400.2.f.be 4 60.l odd 4 2
8640.2.a.cn 2 8.b even 2 1
8640.2.a.cq 2 8.d odd 2 1
8640.2.a.db 2 24.h odd 2 1
8640.2.a.de 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{2} + T_{7} - 18 \)
\( T_{11}^{2} - T_{11} - 18 \)
\( T_{13} - 3 \)
\( T_{17}^{2} - 5 T_{17} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -18 + T + T^{2} \)
$11$ \( -18 - T + T^{2} \)
$13$ \( ( -3 + T )^{2} \)
$17$ \( -12 - 5 T + T^{2} \)
$19$ \( -16 + 3 T + T^{2} \)
$23$ \( -6 - 7 T + T^{2} \)
$29$ \( -18 - T + T^{2} \)
$31$ \( 24 + 13 T + T^{2} \)
$37$ \( -6 - 7 T + T^{2} \)
$41$ \( -72 - 2 T + T^{2} \)
$43$ \( -12 + 5 T + T^{2} \)
$47$ \( -16 - 3 T + T^{2} \)
$53$ \( -72 - 2 T + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( -6 + 7 T + T^{2} \)
$67$ \( 92 + 21 T + T^{2} \)
$71$ \( -48 - 10 T + T^{2} \)
$73$ \( -18 - T + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( ( 8 + T )^{2} \)
$97$ \( -134 + 11 T + T^{2} \)
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