Properties

Label 6720.2.a.x
Level $6720$
Weight $2$
Character orbit 6720.a
Self dual yes
Analytic conductor $53.659$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6594701583\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{5} + q^{7} + q^{9} - 6 q^{11} + 4 q^{13} - q^{15} + 6 q^{17} - 2 q^{19} - q^{21} + q^{25} - q^{27} - 6 q^{29} - 10 q^{31} + 6 q^{33} + q^{35} - 2 q^{37} - 4 q^{39} - 6 q^{41} + 4 q^{43} + q^{45} + q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 14 q^{61} + q^{63} + 4 q^{65} + 4 q^{67} + 6 q^{71} - 4 q^{73} - q^{75} - 6 q^{77} - 16 q^{79} + q^{81} + 12 q^{83} + 6 q^{85} + 6 q^{87} + 6 q^{89} + 4 q^{91} + 10 q^{93} - 2 q^{95} - 16 q^{97} - 6 q^{99} + O(q^{100}) \)

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
0
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
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 6720.2.a.x 1
4.b odd 2 1 6720.2.a.ch 1
8.b even 2 1 420.2.a.c 1
8.d odd 2 1 1680.2.a.a 1
24.f even 2 1 5040.2.a.bc 1
24.h odd 2 1 1260.2.a.i 1
40.e odd 2 1 8400.2.a.cj 1
40.f even 2 1 2100.2.a.d 1
40.i odd 4 2 2100.2.k.j 2
56.h odd 2 1 2940.2.a.f 1
56.j odd 6 2 2940.2.q.i 2
56.p even 6 2 2940.2.q.e 2
120.i odd 2 1 6300.2.a.a 1
120.w even 4 2 6300.2.k.a 2
168.i even 2 1 8820.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 8.b even 2 1
1260.2.a.i 1 24.h odd 2 1
1680.2.a.a 1 8.d odd 2 1
2100.2.a.d 1 40.f even 2 1
2100.2.k.j 2 40.i odd 4 2
2940.2.a.f 1 56.h odd 2 1
2940.2.q.e 2 56.p even 6 2
2940.2.q.i 2 56.j odd 6 2
5040.2.a.bc 1 24.f even 2 1
6300.2.a.a 1 120.i odd 2 1
6300.2.k.a 2 120.w even 4 2
6720.2.a.x 1 1.a even 1 1 trivial
6720.2.a.ch 1 4.b odd 2 1
8400.2.a.cj 1 40.e odd 2 1
8820.2.a.b 1 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6720))\):

\( T_{11} + 6 \)
\( T_{13} - 4 \)
\( T_{17} - 6 \)
\( T_{19} + 2 \)
\( T_{23} \)
\( T_{29} + 6 \)
\( T_{31} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( 6 + T \)
$13$ \( -4 + T \)
$17$ \( -6 + T \)
$19$ \( 2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 10 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( T \)
$53$ \( -12 + T \)
$59$ \( T \)
$61$ \( 14 + T \)
$67$ \( -4 + T \)
$71$ \( -6 + T \)
$73$ \( 4 + T \)
$79$ \( 16 + T \)
$83$ \( -12 + T \)
$89$ \( -6 + T \)
$97$ \( 16 + T \)
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