Properties

Label 2760.2.a.q
Level $2760$
Weight $2$
Character orbit 2760.a
Self dual yes
Analytic conductor $22.039$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} - q^{7} + q^{9} + (3 \beta - 2) q^{11} + ( - \beta - 4) q^{13} + q^{15} + ( - \beta - 3) q^{17} - 5 \beta q^{19} - q^{21} - q^{23} + q^{25} + q^{27} + (3 \beta - 3) q^{29} + ( - 4 \beta - 5) q^{31} + (3 \beta - 2) q^{33} - q^{35} - 3 q^{37} + ( - \beta - 4) q^{39} + (\beta - 3) q^{41} + 2 q^{43} + q^{45} + (\beta + 4) q^{47} - 6 q^{49} + ( - \beta - 3) q^{51} + ( - \beta - 5) q^{53} + (3 \beta - 2) q^{55} - 5 \beta q^{57} + ( - 5 \beta - 5) q^{59} + (5 \beta - 6) q^{61} - q^{63} + ( - \beta - 4) q^{65} + (4 \beta + 5) q^{67} - q^{69} + (5 \beta - 5) q^{71} + 11 \beta q^{73} + q^{75} + ( - 3 \beta + 2) q^{77} + 4 \beta q^{79} + q^{81} + (5 \beta + 9) q^{83} + ( - \beta - 3) q^{85} + (3 \beta - 3) q^{87} + ( - 2 \beta - 8) q^{89} + (\beta + 4) q^{91} + ( - 4 \beta - 5) q^{93} - 5 \beta q^{95} + (2 \beta - 4) q^{97} + (3 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 8 q^{13} + 2 q^{15} - 6 q^{17} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} - 10 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} - 8 q^{39} - 6 q^{41} + 4 q^{43} + 2 q^{45} + 8 q^{47} - 12 q^{49} - 6 q^{51} - 10 q^{53} - 4 q^{55} - 10 q^{59} - 12 q^{61} - 2 q^{63} - 8 q^{65} + 10 q^{67} - 2 q^{69} - 10 q^{71} + 2 q^{75} + 4 q^{77} + 2 q^{81} + 18 q^{83} - 6 q^{85} - 6 q^{87} - 16 q^{89} + 8 q^{91} - 10 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 2760.2.a.q 2
3.b odd 2 1 8280.2.a.y 2
4.b odd 2 1 5520.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.q 2 1.a even 1 1 trivial
5520.2.a.bk 2 4.b odd 2 1
8280.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2760))\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 14 \) 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
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$19$ \( T^{2} - 50 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T - 7 \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 14 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T - 25 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T - 14 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T - 25 \) Copy content Toggle raw display
$73$ \( T^{2} - 242 \) Copy content Toggle raw display
$79$ \( T^{2} - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 18T + 31 \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
show more
show less