Properties

Label 2280.2.a.s
Level $2280$
Weight $2$
Character orbit 2280.a
Self dual yes
Analytic conductor $18.206$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + ( -1 + \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} + ( -1 + \beta_{1} ) q^{7} + q^{9} + ( -1 - \beta_{2} ) q^{11} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{13} - q^{15} + ( -2 - \beta_{1} - \beta_{2} ) q^{17} - q^{19} + ( 1 - \beta_{1} ) q^{21} + ( -4 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} - q^{27} + ( 1 - \beta_{2} ) q^{29} + ( -\beta_{1} + 3 \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{33} + ( -1 + \beta_{1} ) q^{35} + ( -3 + \beta_{1} ) q^{37} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{43} + q^{45} + ( -4 + \beta_{1} + \beta_{2} ) q^{47} + ( 3 + \beta_{1} - \beta_{2} ) q^{49} + ( 2 + \beta_{1} + \beta_{2} ) q^{51} + ( 2 + \beta_{1} + \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + q^{57} + ( -6 - 2 \beta_{2} ) q^{59} + ( -4 + 3 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -1 + \beta_{1} ) q^{63} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{65} + 4 \beta_{2} q^{67} + ( 4 + \beta_{1} + \beta_{2} ) q^{69} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} - q^{75} + ( -\beta_{1} + 3 \beta_{2} ) q^{77} + q^{81} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{83} + ( -2 - \beta_{1} - \beta_{2} ) q^{85} + ( -1 + \beta_{2} ) q^{87} + ( -1 + 2 \beta_{1} - 5 \beta_{2} ) q^{89} + ( -8 - \beta_{1} - 5 \beta_{2} ) q^{91} + ( \beta_{1} - 3 \beta_{2} ) q^{93} - q^{95} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{97} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{15} - 6 q^{17} - 3 q^{19} + 2 q^{21} - 12 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} - 8 q^{37} - 14 q^{43} + 3 q^{45} - 12 q^{47} + 11 q^{49} + 6 q^{51} + 6 q^{53} - 2 q^{55} + 3 q^{57} - 16 q^{59} - 6 q^{61} - 2 q^{63} - 4 q^{67} + 12 q^{69} - 12 q^{71} - 2 q^{73} - 3 q^{75} - 4 q^{77} + 3 q^{81} - 4 q^{83} - 6 q^{85} - 4 q^{87} + 4 q^{89} - 20 q^{91} + 4 q^{93} - 3 q^{95} - 4 q^{97} - 2 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
0.470683
−1.81361
2.34292
0 −1.00000 0 1.00000 0 −3.30777 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.52444 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 3.83221 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.s 3
3.b odd 2 1 6840.2.a.bf 3
4.b odd 2 1 4560.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.s 3 1.a even 1 1 trivial
4560.2.a.bv 3 4.b odd 2 1
6840.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\):

\( T_{7}^{3} + 2 T_{7}^{2} - 14 T_{7} - 32 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8 \)
\( T_{13}^{3} - 34 T_{13} - 76 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -32 - 14 T + 2 T^{2} + T^{3} \)
$11$ \( -8 - 6 T + 2 T^{2} + T^{3} \)
$13$ \( -76 - 34 T + T^{3} \)
$17$ \( -64 - 16 T + 6 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -64 + 20 T + 12 T^{2} + T^{3} \)
$29$ \( 4 - 2 T - 4 T^{2} + T^{3} \)
$31$ \( -256 - 60 T + 4 T^{2} + T^{3} \)
$37$ \( -44 + 6 T + 8 T^{2} + T^{3} \)
$41$ \( 124 - 58 T + T^{3} \)
$43$ \( -296 + 10 T + 14 T^{2} + T^{3} \)
$47$ \( -32 + 20 T + 12 T^{2} + T^{3} \)
$53$ \( 64 - 16 T - 6 T^{2} + T^{3} \)
$59$ \( -32 + 56 T + 16 T^{2} + T^{3} \)
$61$ \( -176 - 144 T + 6 T^{2} + T^{3} \)
$67$ \( 64 - 112 T + 4 T^{2} + T^{3} \)
$71$ \( -1088 - 88 T + 12 T^{2} + T^{3} \)
$73$ \( -8 - 68 T + 2 T^{2} + T^{3} \)
$79$ \( T^{3} \)
$83$ \( -688 - 156 T + 4 T^{2} + T^{3} \)
$89$ \( 1228 - 186 T - 4 T^{2} + T^{3} \)
$97$ \( 124 - 106 T + 4 T^{2} + T^{3} \)
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