Properties

Label 2280.2.a.t
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: 3.3.1016.1
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_{2} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + \beta_{2} q^{7} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} + ( 2 + \beta_{2} ) q^{13} - q^{15} + ( 1 + \beta_{1} ) q^{17} - q^{19} + \beta_{2} q^{21} + ( 1 - \beta_{1} ) q^{23} + q^{25} + q^{27} + ( 1 + \beta_{1} - \beta_{2} ) q^{29} + ( 3 + \beta_{1} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{33} -\beta_{2} q^{35} + ( 2 + \beta_{2} ) q^{37} + ( 2 + \beta_{2} ) q^{39} + ( -1 - \beta_{1} - \beta_{2} ) q^{41} + ( 4 + \beta_{2} ) q^{43} - q^{45} + ( -1 + \beta_{1} ) q^{47} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{49} + ( 1 + \beta_{1} ) q^{51} + ( 1 + \beta_{1} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} ) q^{55} - q^{57} + ( 4 + 2 \beta_{2} ) q^{59} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{61} + \beta_{2} q^{63} + ( -2 - \beta_{2} ) q^{65} + ( 2 + 2 \beta_{1} ) q^{67} + ( 1 - \beta_{1} ) q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + 10 q^{73} + q^{75} + ( -7 - \beta_{1} + 4 \beta_{2} ) q^{77} + ( -2 - 2 \beta_{1} ) q^{79} + q^{81} + ( 5 + 3 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 - \beta_{1} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} ) q^{87} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{89} + ( 9 - \beta_{1} ) q^{91} + ( 3 + \beta_{1} ) q^{93} + q^{95} + ( 10 + \beta_{2} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + 4 q^{11} + 6 q^{13} - 3 q^{15} + 2 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 6 q^{37} + 6 q^{39} - 2 q^{41} + 12 q^{43} - 3 q^{45} - 4 q^{47} + 7 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{55} - 3 q^{57} + 12 q^{59} + 14 q^{61} - 6 q^{65} + 4 q^{67} + 4 q^{69} + 8 q^{71} + 30 q^{73} + 3 q^{75} - 20 q^{77} - 4 q^{79} + 3 q^{81} + 12 q^{83} - 2 q^{85} + 2 q^{87} - 2 q^{89} + 28 q^{91} + 8 q^{93} + 3 q^{95} + 30 q^{97} + 4 q^{99} + O(q^{100}) \)

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

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

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
0.321637
2.85577
−2.17741
0 1.00000 0 −1.00000 0 −4.21819 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 1.29966 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 2.91852 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.t 3
3.b odd 2 1 6840.2.a.bn 3
4.b odd 2 1 4560.2.a.bq 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.t 3 1.a even 1 1 trivial
4560.2.a.bq 3 4.b odd 2 1
6840.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\):

\( T_{7}^{3} - 14 T_{7} + 16 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 26 T_{11} + 96 \)
\( T_{13}^{3} - 6 T_{13}^{2} - 2 T_{13} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 16 - 14 T + T^{3} \)
$11$ \( 96 - 26 T - 4 T^{2} + T^{3} \)
$13$ \( 36 - 2 T - 6 T^{2} + T^{3} \)
$17$ \( 16 - 24 T - 2 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 32 - 20 T - 4 T^{2} + T^{3} \)
$29$ \( 156 - 46 T - 2 T^{2} + T^{3} \)
$31$ \( 48 - 4 T - 8 T^{2} + T^{3} \)
$37$ \( 36 - 2 T - 6 T^{2} + T^{3} \)
$41$ \( 36 - 30 T + 2 T^{2} + T^{3} \)
$43$ \( 8 + 34 T - 12 T^{2} + T^{3} \)
$47$ \( -32 - 20 T + 4 T^{2} + T^{3} \)
$53$ \( 16 - 24 T - 2 T^{2} + T^{3} \)
$59$ \( 288 - 8 T - 12 T^{2} + T^{3} \)
$61$ \( 576 - 32 T - 14 T^{2} + T^{3} \)
$67$ \( 128 - 96 T - 4 T^{2} + T^{3} \)
$71$ \( 768 - 104 T - 8 T^{2} + T^{3} \)
$73$ \( ( -10 + T )^{3} \)
$79$ \( -128 - 96 T + 4 T^{2} + T^{3} \)
$83$ \( 3456 - 284 T - 12 T^{2} + T^{3} \)
$89$ \( -508 - 126 T + 2 T^{2} + T^{3} \)
$97$ \( -844 + 286 T - 30 T^{2} + T^{3} \)
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