Properties

Label 2280.2.a.q
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + (\beta - 3) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + (\beta - 3) q^{7} + q^{9} + ( - 3 \beta - 1) q^{11} + ( - \beta - 3) q^{13} + q^{15} + 4 \beta q^{17} + q^{19} + (\beta - 3) q^{21} - 6 q^{23} + q^{25} + q^{27} + (3 \beta - 5) q^{29} - 2 q^{31} + ( - 3 \beta - 1) q^{33} + (\beta - 3) q^{35} + ( - \beta + 1) q^{37} + ( - \beta - 3) q^{39} + ( - 5 \beta + 3) q^{41} + (\beta - 3) q^{43} + q^{45} - 10 q^{47} + ( - 6 \beta + 5) q^{49} + 4 \beta q^{51} + ( - 3 \beta - 1) q^{55} + q^{57} + (2 \beta - 6) q^{59} + ( - 2 \beta + 2) q^{61} + (\beta - 3) q^{63} + ( - \beta - 3) q^{65} + (4 \beta - 8) q^{67} - 6 q^{69} + ( - 6 \beta + 2) q^{71} + ( - 4 \beta - 6) q^{73} + q^{75} + (8 \beta - 6) q^{77} - 8 q^{79} + q^{81} + (2 \beta + 12) q^{83} + 4 \beta q^{85} + (3 \beta - 5) q^{87} + (5 \beta + 1) q^{89} + 6 q^{91} - 2 q^{93} + q^{95} + ( - 5 \beta - 7) q^{97} + ( - 3 \beta - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} + 2 q^{19} - 6 q^{21} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} - 20 q^{47} + 10 q^{49} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 4 q^{61} - 6 q^{63} - 6 q^{65} - 16 q^{67} - 12 q^{69} + 4 q^{71} - 12 q^{73} + 2 q^{75} - 12 q^{77} - 16 q^{79} + 2 q^{81} + 24 q^{83} - 10 q^{87} + 2 q^{89} + 12 q^{91} - 4 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−1.73205
1.73205
0 1.00000 0 1.00000 0 −4.73205 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −1.26795 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.q 2
3.b odd 2 1 6840.2.a.v 2
4.b odd 2 1 4560.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.q 2 1.a even 1 1 trivial
4560.2.a.bi 2 4.b odd 2 1
6840.2.a.v 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2280))\):

\( T_{7}^{2} + 6T_{7} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 26 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$17$ \( T^{2} - 48 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 10T - 2 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 66 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 24T + 132 \) Copy content Toggle raw display
$89$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T - 26 \) Copy content Toggle raw display
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