Properties

Label 2280.2.a.r
Level $2280$
Weight $2$
Character orbit 2280.a
Self dual yes
Analytic conductor $18.206$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.2058916609\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} + ( 1 - \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} + q^{15} + 2 \beta_{1} q^{17} + q^{19} + ( 1 + \beta_{2} ) q^{21} + ( -3 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} - q^{27} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( -1 + \beta_{2} ) q^{33} + ( 1 + \beta_{2} ) q^{35} + ( -2 - 3 \beta_{1} ) q^{37} + ( 2 - \beta_{1} ) q^{39} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{41} + 3 \beta_{1} q^{43} - q^{45} + ( -5 + \beta_{1} + \beta_{2} ) q^{47} + ( 6 + \beta_{1} - \beta_{2} ) q^{49} -2 \beta_{1} q^{51} + ( 7 - \beta_{1} - \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{55} - q^{57} + ( 7 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( 2 - \beta_{1} ) q^{65} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 3 + \beta_{1} + \beta_{2} ) q^{69} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} - q^{75} + ( 11 + \beta_{1} - 3 \beta_{2} ) q^{77} + ( -3 - \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{83} -2 \beta_{1} q^{85} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{87} + ( 8 + 3 \beta_{1} ) q^{89} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{91} + ( 2 + 2 \beta_{1} ) q^{93} - q^{95} + ( -2 - 3 \beta_{1} ) q^{97} + ( 1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 3q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 3q^{5} - 2q^{7} + 3q^{9} + 4q^{11} - 6q^{13} + 3q^{15} + 3q^{19} + 2q^{21} - 8q^{23} + 3q^{25} - 3q^{27} + 10q^{29} - 6q^{31} - 4q^{33} + 2q^{35} - 6q^{37} + 6q^{39} + 4q^{41} - 3q^{45} - 16q^{47} + 19q^{49} + 22q^{53} - 4q^{55} - 3q^{57} + 22q^{59} + 2q^{61} - 2q^{63} + 6q^{65} + 4q^{67} + 8q^{69} - 8q^{71} + 10q^{73} - 3q^{75} + 36q^{77} - 10q^{79} + 3q^{81} + 2q^{83} - 10q^{87} + 24q^{89} + 8q^{91} + 6q^{93} - 3q^{95} - 6q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1} + 9\)\()/2\)

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
3.12489
−0.363328
−1.76156
0 −1.00000 0 −1.00000 0 −3.60975 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.77801 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 4.38776 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(19\) \(-1\)

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 2280.2.a.r 3
3.b odd 2 1 6840.2.a.bk 3
4.b odd 2 1 4560.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.r 3 1.a even 1 1 trivial
4560.2.a.bt 3 4.b odd 2 1
6840.2.a.bk 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\):

\( T_{7}^{3} + 2 T_{7}^{2} - 18 T_{7} - 44 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 14 T_{11} - 8 \)
\( T_{13}^{3} + 6 T_{13}^{2} + 2 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( -44 - 18 T + 2 T^{2} + T^{3} \)
$11$ \( -8 - 14 T - 4 T^{2} + T^{3} \)
$13$ \( -4 + 2 T + 6 T^{2} + T^{3} \)
$17$ \( 64 - 40 T + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -16 - 4 T + 8 T^{2} + T^{3} \)
$29$ \( 268 - 18 T - 10 T^{2} + T^{3} \)
$31$ \( -136 - 28 T + 6 T^{2} + T^{3} \)
$37$ \( -388 - 78 T + 6 T^{2} + T^{3} \)
$41$ \( 400 - 90 T - 4 T^{2} + T^{3} \)
$43$ \( 216 - 90 T + T^{3} \)
$47$ \( -16 + 60 T + 16 T^{2} + T^{3} \)
$53$ \( -176 + 136 T - 22 T^{2} + T^{3} \)
$59$ \( 976 + 40 T - 22 T^{2} + T^{3} \)
$61$ \( -160 - 96 T - 2 T^{2} + T^{3} \)
$67$ \( -128 - 96 T - 4 T^{2} + T^{3} \)
$71$ \( -1600 - 184 T + 8 T^{2} + T^{3} \)
$73$ \( 488 - 100 T - 10 T^{2} + T^{3} \)
$79$ \( -64 + 10 T^{2} + T^{3} \)
$83$ \( 328 - 100 T - 2 T^{2} + T^{3} \)
$89$ \( 424 + 102 T - 24 T^{2} + T^{3} \)
$97$ \( -388 - 78 T + 6 T^{2} + T^{3} \)
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