Properties

Label 3240.2.a.n
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + \beta q^{7} +O(q^{10})\) \( q + q^{5} + \beta q^{7} + ( 3 - \beta ) q^{11} + ( 2 + \beta ) q^{13} + 2 q^{17} + q^{19} + ( -4 + \beta ) q^{23} + q^{25} + ( 3 - \beta ) q^{29} + ( 3 - \beta ) q^{31} + \beta q^{35} + ( -4 + 2 \beta ) q^{37} + ( 1 - 2 \beta ) q^{41} + 4 q^{43} + ( 2 + \beta ) q^{47} + ( 7 + \beta ) q^{49} + ( 4 + \beta ) q^{53} + ( 3 - \beta ) q^{55} -13 q^{59} + ( 2 - 2 \beta ) q^{61} + ( 2 + \beta ) q^{65} + ( -6 - 2 \beta ) q^{67} + ( -5 + \beta ) q^{71} + ( 2 - 4 \beta ) q^{73} + ( -14 + 2 \beta ) q^{77} + ( 4 - 2 \beta ) q^{79} + ( 4 + 2 \beta ) q^{83} + 2 q^{85} + ( 3 - 3 \beta ) q^{89} + ( 14 + 3 \beta ) q^{91} + q^{95} + 16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + q^{7} + 5 q^{11} + 5 q^{13} + 4 q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 5 q^{29} + 5 q^{31} + q^{35} - 6 q^{37} + 8 q^{43} + 5 q^{47} + 15 q^{49} + 9 q^{53} + 5 q^{55} - 26 q^{59} + 2 q^{61} + 5 q^{65} - 14 q^{67} - 9 q^{71} - 26 q^{77} + 6 q^{79} + 10 q^{83} + 4 q^{85} + 3 q^{89} + 31 q^{91} + 2 q^{95} + 32 q^{97} + O(q^{100}) \)

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
−3.27492
4.27492
0 0 0 1.00000 0 −3.27492 0 0 0
1.2 0 0 0 1.00000 0 4.27492 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.n yes 2
3.b odd 2 1 3240.2.a.h 2
4.b odd 2 1 6480.2.a.bm 2
9.c even 3 2 3240.2.q.y 4
9.d odd 6 2 3240.2.q.be 4
12.b even 2 1 6480.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.h 2 3.b odd 2 1
3240.2.a.n yes 2 1.a even 1 1 trivial
3240.2.q.y 4 9.c even 3 2
3240.2.q.be 4 9.d odd 6 2
6480.2.a.bf 2 12.b even 2 1
6480.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3240))\):

\( T_{7}^{2} - T_{7} - 14 \)
\( T_{11}^{2} - 5 T_{11} - 8 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -14 - T + T^{2} \)
$11$ \( -8 - 5 T + T^{2} \)
$13$ \( -8 - 5 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -2 + 7 T + T^{2} \)
$29$ \( -8 - 5 T + T^{2} \)
$31$ \( -8 - 5 T + T^{2} \)
$37$ \( -48 + 6 T + T^{2} \)
$41$ \( -57 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -8 - 5 T + T^{2} \)
$53$ \( 6 - 9 T + T^{2} \)
$59$ \( ( 13 + T )^{2} \)
$61$ \( -56 - 2 T + T^{2} \)
$67$ \( -8 + 14 T + T^{2} \)
$71$ \( 6 + 9 T + T^{2} \)
$73$ \( -228 + T^{2} \)
$79$ \( -48 - 6 T + T^{2} \)
$83$ \( -32 - 10 T + T^{2} \)
$89$ \( -126 - 3 T + T^{2} \)
$97$ \( ( -16 + T )^{2} \)
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