Properties

Label 2880.2.a.bi
Level $2880$
Weight $2$
Character orbit 2880.a
Self dual yes
Analytic conductor $22.997$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.9969157821\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} -\beta q^{7} +O(q^{10})\) \( q - q^{5} -\beta q^{7} -\beta q^{11} -4 q^{13} + 2 q^{17} -2 \beta q^{23} + q^{25} + 6 q^{29} + 2 \beta q^{31} + \beta q^{35} -8 q^{37} + 8 q^{41} + 2 \beta q^{47} + 13 q^{49} + 6 q^{53} + \beta q^{55} -\beta q^{59} -10 q^{61} + 4 q^{65} -2 \beta q^{67} + 2 \beta q^{71} + 6 q^{73} + 20 q^{77} -2 \beta q^{79} + 2 \beta q^{83} -2 q^{85} + 4 q^{89} + 4 \beta q^{91} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} - 8q^{13} + 4q^{17} + 2q^{25} + 12q^{29} - 16q^{37} + 16q^{41} + 26q^{49} + 12q^{53} - 20q^{61} + 8q^{65} + 12q^{73} + 40q^{77} - 4q^{85} + 8q^{89} + 4q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
0 0 0 −1.00000 0 −4.47214 0 0 0
1.2 0 0 0 −1.00000 0 4.47214 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.a.bi 2
3.b odd 2 1 2880.2.a.bj 2
4.b odd 2 1 inner 2880.2.a.bi 2
8.b even 2 1 1440.2.a.q yes 2
8.d odd 2 1 1440.2.a.q yes 2
12.b even 2 1 2880.2.a.bj 2
24.f even 2 1 1440.2.a.p 2
24.h odd 2 1 1440.2.a.p 2
40.e odd 2 1 7200.2.a.cg 2
40.f even 2 1 7200.2.a.cg 2
40.i odd 4 2 7200.2.f.bj 4
40.k even 4 2 7200.2.f.bj 4
120.i odd 2 1 7200.2.a.ch 2
120.m even 2 1 7200.2.a.ch 2
120.q odd 4 2 7200.2.f.be 4
120.w even 4 2 7200.2.f.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.a.p 2 24.f even 2 1
1440.2.a.p 2 24.h odd 2 1
1440.2.a.q yes 2 8.b even 2 1
1440.2.a.q yes 2 8.d odd 2 1
2880.2.a.bi 2 1.a even 1 1 trivial
2880.2.a.bi 2 4.b odd 2 1 inner
2880.2.a.bj 2 3.b odd 2 1
2880.2.a.bj 2 12.b even 2 1
7200.2.a.cg 2 40.e odd 2 1
7200.2.a.cg 2 40.f even 2 1
7200.2.a.ch 2 120.i odd 2 1
7200.2.a.ch 2 120.m even 2 1
7200.2.f.be 4 120.q odd 4 2
7200.2.f.be 4 120.w even 4 2
7200.2.f.bj 4 40.i odd 4 2
7200.2.f.bj 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2880))\):

\( T_{7}^{2} - 20 \)
\( T_{11}^{2} - 20 \)
\( T_{13} + 4 \)
\( T_{17} - 2 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -20 + T^{2} \)
$11$ \( -20 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( -80 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -80 + T^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -80 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -20 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( -80 + T^{2} \)
$71$ \( -80 + T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( -80 + T^{2} \)
$89$ \( ( -4 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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