Properties

Label 6720.2.a.cp
Level $6720$
Weight $2$
Character orbit 6720.a
Self dual yes
Analytic conductor $53.659$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3360)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + q^{7} + q^{9} + \beta q^{13} + q^{15} - \beta q^{17} - q^{21} + (\beta + 2) q^{23} + q^{25} - q^{27} + ( - \beta - 4) q^{29} + ( - \beta - 2) q^{31} - q^{35} + ( - \beta - 4) q^{37} - \beta q^{39} + (2 \beta + 2) q^{41} + ( - \beta - 6) q^{43} - q^{45} + ( - \beta + 2) q^{47} + q^{49} + \beta q^{51} - 2 q^{53} + 4 q^{59} + ( - \beta - 8) q^{61} + q^{63} - \beta q^{65} + ( - \beta + 2) q^{67} + ( - \beta - 2) q^{69} + (3 \beta + 2) q^{71} + (\beta + 4) q^{73} - q^{75} + (2 \beta + 4) q^{79} + q^{81} + (2 \beta + 8) q^{83} + \beta q^{85} + (\beta + 4) q^{87} + ( - 2 \beta + 2) q^{89} + \beta q^{91} + (\beta + 2) q^{93} - \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} - 16 q^{61} + 2 q^{63} + 4 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{87} + 4 q^{89} + 4 q^{93}+O(q^{100}) \) 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
−0.618034
1.61803
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 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.cp 2
4.b odd 2 1 6720.2.a.cv 2
8.b even 2 1 3360.2.a.bh yes 2
8.d odd 2 1 3360.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bd 2 8.d odd 2 1
3360.2.a.bh yes 2 8.b even 2 1
6720.2.a.cp 2 1.a even 1 1 trivial
6720.2.a.cv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6720))\):

\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 16 \) Copy content Toggle raw display
\( T_{29}^{2} + 8T_{29} - 4 \) Copy content Toggle raw display
\( T_{31}^{2} + 4T_{31} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 176 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 16T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$97$ \( T^{2} - 20 \) Copy content Toggle raw display
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