Properties

Label 3240.2.a.p
Level $3240$
Weight $2$
Character orbit 3240.a
Self dual yes
Analytic conductor $25.872$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + ( 2 + \beta ) q^{7} +O(q^{10})\) \( q + q^{5} + ( 2 + \beta ) q^{7} + ( -2 - 2 \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} + ( -4 - 2 \beta ) q^{17} -2 q^{19} + ( -2 + \beta ) q^{23} + q^{25} + ( -5 - 2 \beta ) q^{29} -2 q^{31} + ( 2 + \beta ) q^{35} -6 \beta q^{37} + ( -3 + 4 \beta ) q^{41} + ( -8 - 2 \beta ) q^{43} + ( -2 - \beta ) q^{47} + 4 \beta q^{49} -6 q^{53} + ( -2 - 2 \beta ) q^{55} + ( -4 + 6 \beta ) q^{59} + ( -5 + 2 \beta ) q^{61} + ( -2 + 2 \beta ) q^{65} + ( 8 + \beta ) q^{67} + ( -6 + 2 \beta ) q^{71} -4 \beta q^{73} + ( -10 - 6 \beta ) q^{77} + ( 12 - 2 \beta ) q^{79} + ( 8 - 3 \beta ) q^{83} + ( -4 - 2 \beta ) q^{85} + ( 3 - 4 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} -2 q^{95} + ( -2 + 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 4 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + 4 q^{7} - 4 q^{11} - 4 q^{13} - 8 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 10 q^{29} - 4 q^{31} + 4 q^{35} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 12 q^{53} - 4 q^{55} - 8 q^{59} - 10 q^{61} - 4 q^{65} + 16 q^{67} - 12 q^{71} - 20 q^{77} + 24 q^{79} + 16 q^{83} - 8 q^{85} + 6 q^{89} + 4 q^{91} - 4 q^{95} - 4 q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 0 0 1.00000 0 0.267949 0 0 0
1.2 0 0 0 1.00000 0 3.73205 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.p 2
3.b odd 2 1 3240.2.a.k 2
4.b odd 2 1 6480.2.a.bk 2
9.c even 3 2 1080.2.q.b 4
9.d odd 6 2 360.2.q.b 4
12.b even 2 1 6480.2.a.ba 2
36.f odd 6 2 2160.2.q.h 4
36.h even 6 2 720.2.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.b 4 9.d odd 6 2
720.2.q.h 4 36.h even 6 2
1080.2.q.b 4 9.c even 3 2
2160.2.q.h 4 36.f odd 6 2
3240.2.a.k 2 3.b odd 2 1
3240.2.a.p 2 1.a even 1 1 trivial
6480.2.a.ba 2 12.b even 2 1
6480.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3240))\):

\( T_{7}^{2} - 4 T_{7} + 1 \)
\( T_{11}^{2} + 4 T_{11} - 8 \)
\( T_{17}^{2} + 8 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 1 - 4 T + T^{2} \)
$11$ \( -8 + 4 T + T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( 4 + 8 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 1 + 4 T + T^{2} \)
$29$ \( 13 + 10 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( -108 + T^{2} \)
$41$ \( -39 + 6 T + T^{2} \)
$43$ \( 52 + 16 T + T^{2} \)
$47$ \( 1 + 4 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -92 + 8 T + T^{2} \)
$61$ \( 13 + 10 T + T^{2} \)
$67$ \( 61 - 16 T + T^{2} \)
$71$ \( 24 + 12 T + T^{2} \)
$73$ \( -48 + T^{2} \)
$79$ \( 132 - 24 T + T^{2} \)
$83$ \( 37 - 16 T + T^{2} \)
$89$ \( -39 - 6 T + T^{2} \)
$97$ \( -44 + 4 T + T^{2} \)
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