Properties

Label 6720.2.a.d
Level 6720
Weight 2
Character orbit 6720.a
Self dual yes
Analytic conductor 53.659
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{3} - q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - q^{5} - q^{7} + q^{9} - 2q^{11} - 4q^{13} + q^{15} + 2q^{17} + 2q^{19} + q^{21} - 4q^{23} + q^{25} - q^{27} - 6q^{29} + 2q^{31} + 2q^{33} + q^{35} - 10q^{37} + 4q^{39} - 10q^{41} + 12q^{43} - q^{45} + 8q^{47} + q^{49} - 2q^{51} + 2q^{55} - 2q^{57} - 8q^{59} + 2q^{61} - q^{63} + 4q^{65} - 12q^{67} + 4q^{69} + 10q^{71} + 4q^{73} - q^{75} + 2q^{77} + q^{81} - 12q^{83} - 2q^{85} + 6q^{87} + 2q^{89} + 4q^{91} - 2q^{93} - 2q^{95} - 8q^{97} - 2q^{99} + O(q^{100}) \)

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:

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + 2 T + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 - 2 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - 2 T + 31 T^{2} \)
$37$ \( 1 + 10 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 - 12 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 + 8 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 - 10 T + 71 T^{2} \)
$73$ \( 1 - 4 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 - 2 T + 89 T^{2} \)
$97$ \( 1 + 8 T + 97 T^{2} \)
show more
show less