Properties

Label 7920.2.a.bu
Level $7920$
Weight $2$
Character orbit 7920.a
Self dual yes
Analytic conductor $63.242$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7} +O(q^{10})\) \( q - q^{5} + \beta q^{7} + q^{11} + 2 q^{13} + ( -2 + \beta ) q^{17} -\beta q^{19} + 2 \beta q^{23} + q^{25} + ( -2 - 3 \beta ) q^{29} + ( -4 - \beta ) q^{31} -\beta q^{35} + ( 2 - 3 \beta ) q^{37} -2 q^{41} -4 \beta q^{43} + ( -8 - 2 \beta ) q^{47} + ( -3 + \beta ) q^{49} + ( -2 - \beta ) q^{53} - q^{55} + ( -4 + 2 \beta ) q^{59} + ( 10 - 3 \beta ) q^{61} -2 q^{65} + ( 4 - 4 \beta ) q^{67} + ( -4 + 3 \beta ) q^{71} -2 q^{73} + \beta q^{77} + 6 \beta q^{79} + 2 \beta q^{83} + ( 2 - \beta ) q^{85} + ( 10 + \beta ) q^{89} + 2 \beta q^{91} + \beta q^{95} + ( 2 + 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + q^{7} + 2q^{11} + 4q^{13} - 3q^{17} - q^{19} + 2q^{23} + 2q^{25} - 7q^{29} - 9q^{31} - q^{35} + q^{37} - 4q^{41} - 4q^{43} - 18q^{47} - 5q^{49} - 5q^{53} - 2q^{55} - 6q^{59} + 17q^{61} - 4q^{65} + 4q^{67} - 5q^{71} - 4q^{73} + q^{77} + 6q^{79} + 2q^{83} + 3q^{85} + 21q^{89} + 2q^{91} + q^{95} + 6q^{97} + O(q^{100}) \)

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
−1.56155
2.56155
0 0 0 −1.00000 0 −1.56155 0 0 0
1.2 0 0 0 −1.00000 0 2.56155 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 7920.2.a.bu 2
3.b odd 2 1 880.2.a.o 2
4.b odd 2 1 3960.2.a.w 2
12.b even 2 1 440.2.a.e 2
15.d odd 2 1 4400.2.a.bj 2
15.e even 4 2 4400.2.b.t 4
24.f even 2 1 3520.2.a.bp 2
24.h odd 2 1 3520.2.a.bk 2
33.d even 2 1 9680.2.a.bs 2
60.h even 2 1 2200.2.a.s 2
60.l odd 4 2 2200.2.b.i 4
132.d odd 2 1 4840.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 12.b even 2 1
880.2.a.o 2 3.b odd 2 1
2200.2.a.s 2 60.h even 2 1
2200.2.b.i 4 60.l odd 4 2
3520.2.a.bk 2 24.h odd 2 1
3520.2.a.bp 2 24.f even 2 1
3960.2.a.w 2 4.b odd 2 1
4400.2.a.bj 2 15.d odd 2 1
4400.2.b.t 4 15.e even 4 2
4840.2.a.j 2 132.d odd 2 1
7920.2.a.bu 2 1.a even 1 1 trivial
9680.2.a.bs 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7920))\):

\( T_{7}^{2} - T_{7} - 4 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + 3 T_{17} - 2 \)
\( T_{19}^{2} + T_{19} - 4 \)
\( T_{23}^{2} - 2 T_{23} - 16 \)
\( T_{29}^{2} + 7 T_{29} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -2 + 3 T + T^{2} \)
$19$ \( -4 + T + T^{2} \)
$23$ \( -16 - 2 T + T^{2} \)
$29$ \( -26 + 7 T + T^{2} \)
$31$ \( 16 + 9 T + T^{2} \)
$37$ \( -38 - T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -64 + 4 T + T^{2} \)
$47$ \( 64 + 18 T + T^{2} \)
$53$ \( 2 + 5 T + T^{2} \)
$59$ \( -8 + 6 T + T^{2} \)
$61$ \( 34 - 17 T + T^{2} \)
$67$ \( -64 - 4 T + T^{2} \)
$71$ \( -32 + 5 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( -144 - 6 T + T^{2} \)
$83$ \( -16 - 2 T + T^{2} \)
$89$ \( 106 - 21 T + T^{2} \)
$97$ \( -8 - 6 T + T^{2} \)
show more
show less