Properties

Label 2280.2.a.k
Level $2280$
Weight $2$
Character orbit 2280.a
Self dual yes
Analytic conductor $18.206$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + \beta q^{7} + q^{9} + ( -2 + \beta ) q^{11} + ( 2 - \beta ) q^{13} + q^{15} -2 \beta q^{17} + q^{19} -\beta q^{21} + ( -2 - 2 \beta ) q^{23} + q^{25} - q^{27} -5 \beta q^{29} + ( -2 - 2 \beta ) q^{31} + ( 2 - \beta ) q^{33} -\beta q^{35} + ( 2 + 3 \beta ) q^{37} + ( -2 + \beta ) q^{39} + 7 \beta q^{41} + ( 4 + 5 \beta ) q^{43} - q^{45} + ( -6 + 2 \beta ) q^{47} -5 q^{49} + 2 \beta q^{51} + 6 \beta q^{53} + ( 2 - \beta ) q^{55} - q^{57} + ( -4 + 2 \beta ) q^{59} + ( 4 - 4 \beta ) q^{61} + \beta q^{63} + ( -2 + \beta ) q^{65} -8 \beta q^{67} + ( 2 + 2 \beta ) q^{69} + ( -12 - 2 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} - q^{75} + ( 2 - 2 \beta ) q^{77} + ( -8 - 4 \beta ) q^{79} + q^{81} + ( -2 - 4 \beta ) q^{83} + 2 \beta q^{85} + 5 \beta q^{87} + ( -8 - 3 \beta ) q^{89} + ( -2 + 2 \beta ) q^{91} + ( 2 + 2 \beta ) q^{93} - q^{95} + ( 2 + 7 \beta ) q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{31} + 4 q^{33} + 4 q^{37} - 4 q^{39} + 8 q^{43} - 2 q^{45} - 12 q^{47} - 10 q^{49} + 4 q^{55} - 2 q^{57} - 8 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{69} - 24 q^{71} - 12 q^{73} - 2 q^{75} + 4 q^{77} - 16 q^{79} + 2 q^{81} - 4 q^{83} - 16 q^{89} - 4 q^{91} + 4 q^{93} - 2 q^{95} + 4 q^{97} - 4 q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.00000 0 −1.00000 0 −1.41421 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.41421 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.k 2
3.b odd 2 1 6840.2.a.bb 2
4.b odd 2 1 4560.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.k 2 1.a even 1 1 trivial
4560.2.a.bn 2 4.b odd 2 1
6840.2.a.bb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( 2 + 4 T + T^{2} \)
$13$ \( 2 - 4 T + T^{2} \)
$17$ \( -8 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -4 + 4 T + T^{2} \)
$29$ \( -50 + T^{2} \)
$31$ \( -4 + 4 T + T^{2} \)
$37$ \( -14 - 4 T + T^{2} \)
$41$ \( -98 + T^{2} \)
$43$ \( -34 - 8 T + T^{2} \)
$47$ \( 28 + 12 T + T^{2} \)
$53$ \( -72 + T^{2} \)
$59$ \( 8 + 8 T + T^{2} \)
$61$ \( -16 - 8 T + T^{2} \)
$67$ \( -128 + T^{2} \)
$71$ \( 136 + 24 T + T^{2} \)
$73$ \( 4 + 12 T + T^{2} \)
$79$ \( 32 + 16 T + T^{2} \)
$83$ \( -28 + 4 T + T^{2} \)
$89$ \( 46 + 16 T + T^{2} \)
$97$ \( -94 - 4 T + T^{2} \)
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