Properties

Label 7728.2.a.cf
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.17679757.1
Defining polynomial: \(x^{5} - x^{4} - 14 x^{3} + 17 x^{2} + 23 x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} - q^{7} + q^{9} + ( \beta_{1} + \beta_{2} ) q^{11} + ( 1 + \beta_{4} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( 2 - \beta_{1} + \beta_{4} ) q^{17} + ( -2 + \beta_{1} + \beta_{4} ) q^{19} + q^{21} + q^{23} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( -1 + \beta_{1} + \beta_{3} ) q^{29} + ( -\beta_{1} + \beta_{4} ) q^{31} + ( -\beta_{1} - \beta_{2} ) q^{33} + ( 1 + \beta_{1} ) q^{35} + ( 1 - \beta_{1} - \beta_{3} ) q^{37} + ( -1 - \beta_{4} ) q^{39} + ( 1 - \beta_{1} - \beta_{3} ) q^{41} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( 1 + \beta_{3} + \beta_{4} ) q^{47} + q^{49} + ( -2 + \beta_{1} - \beta_{4} ) q^{51} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{53} + ( -2 - 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 2 - \beta_{1} - \beta_{4} ) q^{57} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{59} + \beta_{3} q^{61} - q^{63} + ( -3 - \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{65} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{67} - q^{69} + ( -2 \beta_{1} + \beta_{3} ) q^{71} + ( 3 \beta_{1} + \beta_{2} ) q^{73} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{75} + ( -\beta_{1} - \beta_{2} ) q^{77} + ( -\beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( -6 - 3 \beta_{1} + \beta_{4} ) q^{83} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{85} + ( 1 - \beta_{1} - \beta_{3} ) q^{87} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{89} + ( -1 - \beta_{4} ) q^{91} + ( \beta_{1} - \beta_{4} ) q^{93} + ( -6 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{95} + ( 9 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{97} + ( \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} - 6q^{5} - 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} - 6q^{5} - 5q^{7} + 5q^{9} + 4q^{13} + 6q^{15} + 8q^{17} - 10q^{19} + 5q^{21} + 5q^{23} + 11q^{25} - 5q^{27} - 3q^{29} - 2q^{31} + 6q^{35} + 3q^{37} - 4q^{39} + 3q^{41} - 24q^{43} - 6q^{45} + 5q^{47} + 5q^{49} - 8q^{51} + 9q^{53} - 15q^{55} + 10q^{57} - 3q^{59} + q^{61} - 5q^{63} - 15q^{65} - 9q^{67} - 5q^{69} - q^{71} + 2q^{73} - 11q^{75} + 5q^{81} - 34q^{83} + 9q^{85} + 3q^{87} + 11q^{89} - 4q^{91} + 2q^{93} - 27q^{95} + 47q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 14 x^{3} + 17 x^{2} + 23 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + \nu^{2} - 10 \nu - 2 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 12 \nu^{2} + 7 \nu + 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{3} - \beta_{2} + 10 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 12 \beta_{2} - 7 \beta_{1} + 62\)

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
3.07615
2.26247
0.366085
−1.08402
−3.62069
0 −1.00000 0 −4.07615 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 −3.26247 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 −1.36608 0 −1.00000 0 1.00000 0
1.4 0 −1.00000 0 0.0840175 0 −1.00000 0 1.00000 0
1.5 0 −1.00000 0 2.62069 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.cf 5
4.b odd 2 1 3864.2.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.v 5 4.b odd 2 1
7728.2.a.cf 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{5} + 6 T_{5}^{4} - 43 T_{5}^{2} - 44 T_{5} + 4 \)
\( T_{11}^{5} - 61 T_{11}^{3} + 16 T_{11}^{2} + 832 T_{11} - 1024 \)
\( T_{13}^{5} - 4 T_{13}^{4} - 40 T_{13}^{3} + 95 T_{13}^{2} + 432 T_{13} - 284 \)
\( T_{17}^{5} - 8 T_{17}^{4} - 25 T_{17}^{3} + 240 T_{17}^{2} + 20 T_{17} - 1024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( 4 - 44 T - 43 T^{2} + 6 T^{4} + T^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( -1024 + 832 T + 16 T^{2} - 61 T^{3} + T^{5} \)
$13$ \( -284 + 432 T + 95 T^{2} - 40 T^{3} - 4 T^{4} + T^{5} \)
$17$ \( -1024 + 20 T + 240 T^{2} - 25 T^{3} - 8 T^{4} + T^{5} \)
$19$ \( 4096 + 192 T - 412 T^{2} - 31 T^{3} + 10 T^{4} + T^{5} \)
$23$ \( ( -1 + T )^{5} \)
$29$ \( 10196 + 2512 T - 363 T^{2} - 101 T^{3} + 3 T^{4} + T^{5} \)
$31$ \( -320 + 504 T - 22 T^{2} - 49 T^{3} + 2 T^{4} + T^{5} \)
$37$ \( -10196 + 2512 T + 363 T^{2} - 101 T^{3} - 3 T^{4} + T^{5} \)
$41$ \( -10196 + 2512 T + 363 T^{2} - 101 T^{3} - 3 T^{4} + T^{5} \)
$43$ \( -3904 - 3504 T - 313 T^{2} + 140 T^{3} + 24 T^{4} + T^{5} \)
$47$ \( 256 - 880 T + 748 T^{2} - 122 T^{3} - 5 T^{4} + T^{5} \)
$53$ \( -63568 + 8332 T + 1500 T^{2} - 185 T^{3} - 9 T^{4} + T^{5} \)
$59$ \( 2656 + 3424 T - 250 T^{2} - 159 T^{3} + 3 T^{4} + T^{5} \)
$61$ \( 328 + 2212 T + 34 T^{2} - 101 T^{3} - T^{4} + T^{5} \)
$67$ \( -4352 - 4608 T - 1492 T^{2} - 125 T^{3} + 9 T^{4} + T^{5} \)
$71$ \( 2048 + 6000 T - 10 T^{2} - 181 T^{3} + T^{4} + T^{5} \)
$73$ \( 10832 + 3812 T - 224 T^{2} - 147 T^{3} - 2 T^{4} + T^{5} \)
$79$ \( 1024 + 832 T - 16 T^{2} - 61 T^{3} + T^{5} \)
$83$ \( -11392 - 7600 T + 96 T^{2} + 317 T^{3} + 34 T^{4} + T^{5} \)
$89$ \( -24392 - 5132 T + 2702 T^{2} - 185 T^{3} - 11 T^{4} + T^{5} \)
$97$ \( -4892 - 11572 T - 2313 T^{2} + 687 T^{3} - 47 T^{4} + T^{5} \)
show more
show less