Properties

Label 6720.2.a.cs
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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} - q^{7} + q^{9} + 4 \beta q^{11} + ( 2 - 4 \beta ) q^{13} - q^{15} -2 q^{17} + ( 4 - 4 \beta ) q^{19} + q^{21} -4 q^{23} + q^{25} - q^{27} + 2 q^{29} + ( -8 + 4 \beta ) q^{31} -4 \beta q^{33} - q^{35} + ( -6 + 8 \beta ) q^{37} + ( -2 + 4 \beta ) q^{39} -2 q^{41} + ( -4 + 8 \beta ) q^{43} + q^{45} -8 \beta q^{47} + q^{49} + 2 q^{51} + ( 10 - 4 \beta ) q^{53} + 4 \beta q^{55} + ( -4 + 4 \beta ) q^{57} + ( 4 - 8 \beta ) q^{59} + 2 q^{61} - q^{63} + ( 2 - 4 \beta ) q^{65} -4 q^{67} + 4 q^{69} + ( -12 + 4 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} - q^{75} -4 \beta q^{77} -8 \beta q^{79} + q^{81} + ( -4 - 8 \beta ) q^{83} -2 q^{85} -2 q^{87} -2 q^{89} + ( -2 + 4 \beta ) q^{91} + ( 8 - 4 \beta ) q^{93} + ( 4 - 4 \beta ) q^{95} + ( 2 + 4 \beta ) q^{97} + 4 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + 4q^{11} - 2q^{15} - 4q^{17} + 4q^{19} + 2q^{21} - 8q^{23} + 2q^{25} - 2q^{27} + 4q^{29} - 12q^{31} - 4q^{33} - 2q^{35} - 4q^{37} - 4q^{41} + 2q^{45} - 8q^{47} + 2q^{49} + 4q^{51} + 16q^{53} + 4q^{55} - 4q^{57} + 4q^{61} - 2q^{63} - 8q^{67} + 8q^{69} - 20q^{71} - 16q^{73} - 2q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 16q^{83} - 4q^{85} - 4q^{87} - 4q^{89} + 12q^{93} + 4q^{95} + 8q^{97} + 4q^{99} + O(q^{100}) \)

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
−0.618034
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:

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.cs 2
4.b odd 2 1 6720.2.a.cx 2
8.b even 2 1 1680.2.a.v 2
8.d odd 2 1 105.2.a.b 2
24.f even 2 1 315.2.a.d 2
24.h odd 2 1 5040.2.a.bw 2
40.e odd 2 1 525.2.a.g 2
40.f even 2 1 8400.2.a.cx 2
40.k even 4 2 525.2.d.c 4
56.e even 2 1 735.2.a.k 2
56.k odd 6 2 735.2.i.k 4
56.m even 6 2 735.2.i.i 4
120.m even 2 1 1575.2.a.r 2
120.q odd 4 2 1575.2.d.d 4
168.e odd 2 1 2205.2.a.w 2
280.n even 2 1 3675.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 8.d odd 2 1
315.2.a.d 2 24.f even 2 1
525.2.a.g 2 40.e odd 2 1
525.2.d.c 4 40.k even 4 2
735.2.a.k 2 56.e even 2 1
735.2.i.i 4 56.m even 6 2
735.2.i.k 4 56.k odd 6 2
1575.2.a.r 2 120.m even 2 1
1575.2.d.d 4 120.q odd 4 2
1680.2.a.v 2 8.b even 2 1
2205.2.a.w 2 168.e odd 2 1
3675.2.a.y 2 280.n even 2 1
5040.2.a.bw 2 24.h odd 2 1
6720.2.a.cs 2 1.a even 1 1 trivial
6720.2.a.cx 2 4.b odd 2 1
8400.2.a.cx 2 40.f even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6720))\):

\( T_{11}^{2} - 4 T_{11} - 16 \)
\( T_{13}^{2} - 20 \)
\( T_{17} + 2 \)
\( T_{19}^{2} - 4 T_{19} - 16 \)
\( T_{23} + 4 \)
\( T_{29} - 2 \)
\( T_{31}^{2} + 12 T_{31} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 6 T^{2} + 169 T^{4} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 12 T + 78 T^{2} + 372 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 6 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 16 T + 150 T^{2} - 848 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 38 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 20 T + 222 T^{2} + 1420 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 16 T + 190 T^{2} + 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 16 T + 150 T^{2} + 1328 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 8 T + 190 T^{2} - 776 T^{3} + 9409 T^{4} \)
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