Properties

Label 3240.2.a.o
Level $3240$
Weight $2$
Character orbit 3240.a
Self dual yes
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8715302549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta + 1) q^{7} + 2 q^{17} + ( - 2 \beta - 2) q^{19} + (\beta + 5) q^{23} + q^{25} + (2 \beta - 3) q^{29} + ( - 2 \beta + 6) q^{31} + (\beta + 1) q^{35} - 6 q^{37} + (2 \beta + 5) q^{41} + (4 \beta + 2) q^{43} + (\beta + 7) q^{47} + 2 \beta q^{49} + ( - 4 \beta + 2) q^{53} + ( - 2 \beta + 6) q^{59} - 3 q^{61} + ( - 5 \beta - 1) q^{67} - 4 \beta q^{71} + (4 \beta + 4) q^{73} + ( - 2 \beta - 2) q^{79} + (\beta + 3) q^{83} + 2 q^{85} + ( - 4 \beta + 7) q^{89} + ( - 2 \beta - 2) q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{17} - 4 q^{19} + 10 q^{23} + 2 q^{25} - 6 q^{29} + 12 q^{31} + 2 q^{35} - 12 q^{37} + 10 q^{41} + 4 q^{43} + 14 q^{47} + 4 q^{53} + 12 q^{59} - 6 q^{61} - 2 q^{67} + 8 q^{73} - 4 q^{79} + 6 q^{83} + 4 q^{85} + 14 q^{89} - 4 q^{95} + 4 q^{97}+O(q^{100}) \) 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
−2.44949
2.44949
0 0 0 1.00000 0 −1.44949 0 0 0
1.2 0 0 0 1.00000 0 3.44949 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3240.2.a.o 2
3.b odd 2 1 3240.2.a.j 2
4.b odd 2 1 6480.2.a.bl 2
9.c even 3 2 1080.2.q.c 4
9.d odd 6 2 360.2.q.c 4
12.b even 2 1 6480.2.a.bc 2
36.f odd 6 2 2160.2.q.g 4
36.h even 6 2 720.2.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.c 4 9.d odd 6 2
720.2.q.g 4 36.h even 6 2
1080.2.q.c 4 9.c even 3 2
2160.2.q.g 4 36.f odd 6 2
3240.2.a.j 2 3.b odd 2 1
3240.2.a.o 2 1.a even 1 1 trivial
6480.2.a.bc 2 12.b even 2 1
6480.2.a.bl 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3240))\):

\( T_{7}^{2} - 2T_{7} - 5 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 19 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 15 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 12 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 43 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 12 \) Copy content Toggle raw display
$61$ \( (T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 149 \) Copy content Toggle raw display
$71$ \( T^{2} - 96 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 80 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 3 \) Copy content Toggle raw display
$89$ \( T^{2} - 14T - 47 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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