Properties

Label 7728.2.a.bv
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.1509.1
Defining polynomial: \( x^{3} - x^{2} - 7x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1932)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} - q^{7} + q^{9} + (2 \beta_1 - 1) q^{11} + ( - \beta_{2} + 1) q^{13} - \beta_1 q^{15} + (\beta_{2} - \beta_1 + 1) q^{17} - q^{19} - q^{21} - q^{23} + \beta_{2} q^{25} + q^{27} + ( - \beta_{2} - \beta_1 - 3) q^{29} + (\beta_{2} - \beta_1 - 1) q^{31} + (2 \beta_1 - 1) q^{33} + \beta_1 q^{35} + ( - \beta_{2} + 3 \beta_1 - 3) q^{37} + ( - \beta_{2} + 1) q^{39} + (2 \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - \beta_{2} + 2 \beta_1 - 5) q^{43} - \beta_1 q^{45} + (\beta_{2} + \beta_1 - 4) q^{47} + q^{49} + (\beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{2} - 4) q^{53} + ( - 2 \beta_{2} + \beta_1 - 10) q^{55} - q^{57} + ( - \beta_{2} - 4) q^{59} + (\beta_1 + 7) q^{61} - q^{63} + (\beta_{2} + \beta_1 + 1) q^{65} + \beta_1 q^{67} - q^{69} + (3 \beta_{2} - 2 \beta_1 + 3) q^{71} + (\beta_{2} + \beta_1 + 1) q^{73} + \beta_{2} q^{75} + ( - 2 \beta_1 + 1) q^{77} + (\beta_{2} - 5 \beta_1 - 1) q^{79} + q^{81} + ( - \beta_{2} - \beta_1 - 3) q^{83} + ( - 3 \beta_1 + 4) q^{85} + ( - \beta_{2} - \beta_1 - 3) q^{87} + ( - \beta_{2} + 5) q^{89} + (\beta_{2} - 1) q^{91} + (\beta_{2} - \beta_1 - 1) q^{93} + \beta_1 q^{95} + ( - 3 \beta_{2} + \beta_1 + 3) q^{97} + (2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 3 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 10 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} + 3 q^{39} - q^{41} - 13 q^{43} - q^{45} - 11 q^{47} + 3 q^{49} + 2 q^{51} - 12 q^{53} - 29 q^{55} - 3 q^{57} - 12 q^{59} + 22 q^{61} - 3 q^{63} + 4 q^{65} + q^{67} - 3 q^{69} + 7 q^{71} + 4 q^{73} + q^{77} - 8 q^{79} + 3 q^{81} - 10 q^{83} + 9 q^{85} - 10 q^{87} + 15 q^{89} - 3 q^{91} - 4 q^{93} + q^{95} + 10 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) 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
2.92542
0.551929
−2.47735
0 1.00000 0 −2.92542 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.551929 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.47735 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.bv 3
4.b odd 2 1 1932.2.a.i 3
12.b even 2 1 5796.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.i 3 4.b odd 2 1
5796.2.a.p 3 12.b even 2 1
7728.2.a.bv 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} + T_{5}^{2} - 7T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 29T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 15T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 19T_{17} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 7T - 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 29T + 3 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} - 15 T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 19 T + 32 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} + 3 T - 18 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 15 T - 6 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 57 T + 86 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} - 121 T - 409 \) Copy content Toggle raw display
$43$ \( T^{3} + 13 T^{2} + 19 T - 24 \) Copy content Toggle raw display
$47$ \( T^{3} + 11 T^{2} + 10 T - 108 \) Copy content Toggle raw display
$53$ \( T^{3} + 12 T^{2} + 30 T - 27 \) Copy content Toggle raw display
$59$ \( T^{3} + 12 T^{2} + 30 T - 27 \) Copy content Toggle raw display
$61$ \( T^{3} - 22 T^{2} + 154 T - 339 \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} - 7T + 4 \) Copy content Toggle raw display
$71$ \( T^{3} - 7 T^{2} - 145 T + 1084 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 25 T - 8 \) Copy content Toggle raw display
$79$ \( T^{3} + 8 T^{2} - 155 T - 1278 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} + 3 T - 18 \) Copy content Toggle raw display
$89$ \( T^{3} - 15 T^{2} + 57 T - 54 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} - 121 T - 242 \) Copy content Toggle raw display
show more
show less