Properties

Label 7728.2.a.x
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -2 - \beta ) q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -2 - \beta ) q^{5} + q^{7} + q^{9} + 5 q^{11} + ( 1 - \beta ) q^{13} + ( 2 + \beta ) q^{15} + ( -3 + 2 \beta ) q^{17} + ( -1 - 2 \beta ) q^{19} - q^{21} - q^{23} + ( 2 + 5 \beta ) q^{25} - q^{27} + ( 1 - 4 \beta ) q^{29} -3 q^{31} -5 q^{33} + ( -2 - \beta ) q^{35} -9 q^{37} + ( -1 + \beta ) q^{39} + ( -7 + 4 \beta ) q^{41} + ( 1 + 5 \beta ) q^{43} + ( -2 - \beta ) q^{45} + ( 6 - 2 \beta ) q^{47} + q^{49} + ( 3 - 2 \beta ) q^{51} + ( -6 + 5 \beta ) q^{53} + ( -10 - 5 \beta ) q^{55} + ( 1 + 2 \beta ) q^{57} + 3 \beta q^{59} + ( -5 - 3 \beta ) q^{61} + q^{63} + ( 1 + 2 \beta ) q^{65} + ( 8 - 3 \beta ) q^{67} + q^{69} + ( 3 + 3 \beta ) q^{71} + ( 3 + 4 \beta ) q^{73} + ( -2 - 5 \beta ) q^{75} + 5 q^{77} + q^{79} + q^{81} + ( -3 + 2 \beta ) q^{83} -3 \beta q^{85} + ( -1 + 4 \beta ) q^{87} + ( 7 - 3 \beta ) q^{89} + ( 1 - \beta ) q^{91} + 3 q^{93} + ( 8 + 7 \beta ) q^{95} + ( -15 + 2 \beta ) q^{97} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + 10q^{11} + q^{13} + 5q^{15} - 4q^{17} - 4q^{19} - 2q^{21} - 2q^{23} + 9q^{25} - 2q^{27} - 2q^{29} - 6q^{31} - 10q^{33} - 5q^{35} - 18q^{37} - q^{39} - 10q^{41} + 7q^{43} - 5q^{45} + 10q^{47} + 2q^{49} + 4q^{51} - 7q^{53} - 25q^{55} + 4q^{57} + 3q^{59} - 13q^{61} + 2q^{63} + 4q^{65} + 13q^{67} + 2q^{69} + 9q^{71} + 10q^{73} - 9q^{75} + 10q^{77} + 2q^{79} + 2q^{81} - 4q^{83} - 3q^{85} + 2q^{87} + 11q^{89} + q^{91} + 6q^{93} + 23q^{95} - 28q^{97} + 10q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
0 −1.00000 0 −4.30278 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −0.697224 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 7728.2.a.x 2
4.b odd 2 1 483.2.a.f 2
12.b even 2 1 1449.2.a.j 2
28.d even 2 1 3381.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.f 2 4.b odd 2 1
1449.2.a.j 2 12.b even 2 1
3381.2.a.p 2 28.d even 2 1
7728.2.a.x 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} + 5 T_{5} + 3 \)
\( T_{11} - 5 \)
\( T_{13}^{2} - T_{13} - 3 \)
\( T_{17}^{2} + 4 T_{17} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 3 + 5 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -5 + T )^{2} \)
$13$ \( -3 - T + T^{2} \)
$17$ \( -9 + 4 T + T^{2} \)
$19$ \( -9 + 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -51 + 2 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( ( 9 + T )^{2} \)
$41$ \( -27 + 10 T + T^{2} \)
$43$ \( -69 - 7 T + T^{2} \)
$47$ \( 12 - 10 T + T^{2} \)
$53$ \( -69 + 7 T + T^{2} \)
$59$ \( -27 - 3 T + T^{2} \)
$61$ \( 13 + 13 T + T^{2} \)
$67$ \( 13 - 13 T + T^{2} \)
$71$ \( -9 - 9 T + T^{2} \)
$73$ \( -27 - 10 T + T^{2} \)
$79$ \( ( -1 + T )^{2} \)
$83$ \( -9 + 4 T + T^{2} \)
$89$ \( 1 - 11 T + T^{2} \)
$97$ \( 183 + 28 T + T^{2} \)
show more
show less