Properties

Label 7728.2.a.bl
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: \(\Q(\sqrt{29}) \)
Defining polynomial: \(x^{2} - x - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{29})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} + q^{7} + q^{9} -5 q^{11} + ( 1 - \beta ) q^{13} + \beta q^{15} + q^{17} + ( -1 - 2 \beta ) q^{19} + q^{21} - q^{23} + ( 2 + \beta ) q^{25} + q^{27} -5 q^{29} + ( -1 - 2 \beta ) q^{31} -5 q^{33} + \beta q^{35} + ( 1 - 2 \beta ) q^{37} + ( 1 - \beta ) q^{39} - q^{41} + ( -3 + \beta ) q^{43} + \beta q^{45} + ( -6 + 2 \beta ) q^{47} + q^{49} + q^{51} + ( -4 + 3 \beta ) q^{53} -5 \beta q^{55} + ( -1 - 2 \beta ) q^{57} + ( 6 - 3 \beta ) q^{59} + ( 7 - \beta ) q^{61} + q^{63} -7 q^{65} + \beta q^{67} - q^{69} + ( -7 + 3 \beta ) q^{71} -9 q^{73} + ( 2 + \beta ) q^{75} -5 q^{77} + ( 1 - 4 \beta ) q^{79} + q^{81} + ( 7 - 2 \beta ) q^{83} + \beta q^{85} -5 q^{87} + ( -3 - 3 \beta ) q^{89} + ( 1 - \beta ) q^{91} + ( -1 - 2 \beta ) q^{93} + ( -14 - 3 \beta ) q^{95} + ( -9 + 4 \beta ) q^{97} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} + 2q^{7} + 2q^{9} - 10q^{11} + q^{13} + q^{15} + 2q^{17} - 4q^{19} + 2q^{21} - 2q^{23} + 5q^{25} + 2q^{27} - 10q^{29} - 4q^{31} - 10q^{33} + q^{35} + q^{39} - 2q^{41} - 5q^{43} + q^{45} - 10q^{47} + 2q^{49} + 2q^{51} - 5q^{53} - 5q^{55} - 4q^{57} + 9q^{59} + 13q^{61} + 2q^{63} - 14q^{65} + q^{67} - 2q^{69} - 11q^{71} - 18q^{73} + 5q^{75} - 10q^{77} - 2q^{79} + 2q^{81} + 12q^{83} + q^{85} - 10q^{87} - 9q^{89} + q^{91} - 4q^{93} - 31q^{95} - 14q^{97} - 10q^{99} + O(q^{100}) \)

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
−2.19258
3.19258
0 1.00000 0 −2.19258 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 3.19258 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.bl 2
4.b odd 2 1 3864.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.h 2 4.b odd 2 1
7728.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} - T_{5} - 7 \)
\( T_{11} + 5 \)
\( T_{13}^{2} - T_{13} - 7 \)
\( T_{17} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -7 - T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( -7 - T + T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( -25 + 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( -25 + 4 T + T^{2} \)
$37$ \( -29 + T^{2} \)
$41$ \( ( 1 + T )^{2} \)
$43$ \( -1 + 5 T + T^{2} \)
$47$ \( -4 + 10 T + T^{2} \)
$53$ \( -59 + 5 T + T^{2} \)
$59$ \( -45 - 9 T + T^{2} \)
$61$ \( 35 - 13 T + T^{2} \)
$67$ \( -7 - T + T^{2} \)
$71$ \( -35 + 11 T + T^{2} \)
$73$ \( ( 9 + T )^{2} \)
$79$ \( -115 + 2 T + T^{2} \)
$83$ \( 7 - 12 T + T^{2} \)
$89$ \( -45 + 9 T + T^{2} \)
$97$ \( -67 + 14 T + T^{2} \)
show more
show less