Properties

Label 7728.2.a.bk
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
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 1932)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} - q^{7} + q^{9} + ( -1 + 2 \beta ) q^{11} + ( -3 + \beta ) q^{13} + \beta q^{15} -3 q^{17} + ( 3 + 2 \beta ) q^{19} - q^{21} + q^{23} + ( -2 + \beta ) q^{25} + q^{27} + q^{29} - q^{31} + ( -1 + 2 \beta ) q^{33} -\beta q^{35} + ( 1 - 4 \beta ) q^{37} + ( -3 + \beta ) q^{39} + ( -1 - 2 \beta ) q^{41} + ( 1 + 3 \beta ) q^{43} + \beta q^{45} + ( 6 + 2 \beta ) q^{47} + q^{49} -3 q^{51} + ( 6 + 3 \beta ) q^{53} + ( 6 + \beta ) q^{55} + ( 3 + 2 \beta ) q^{57} + ( -4 - 3 \beta ) q^{59} + ( -3 + 5 \beta ) q^{61} - q^{63} + ( 3 - 2 \beta ) q^{65} + ( 6 - 7 \beta ) q^{67} + q^{69} + ( 5 + 3 \beta ) q^{71} + ( -3 + 6 \beta ) q^{73} + ( -2 + \beta ) q^{75} + ( 1 - 2 \beta ) q^{77} + ( 11 - 2 \beta ) q^{79} + q^{81} + ( 5 - 2 \beta ) q^{83} -3 \beta q^{85} + q^{87} + ( 1 + \beta ) q^{89} + ( 3 - \beta ) q^{91} - q^{93} + ( 6 + 5 \beta ) q^{95} -11 q^{97} + ( -1 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} - 2q^{7} + 2q^{9} - 5q^{13} + q^{15} - 6q^{17} + 8q^{19} - 2q^{21} + 2q^{23} - 3q^{25} + 2q^{27} + 2q^{29} - 2q^{31} - q^{35} - 2q^{37} - 5q^{39} - 4q^{41} + 5q^{43} + q^{45} + 14q^{47} + 2q^{49} - 6q^{51} + 15q^{53} + 13q^{55} + 8q^{57} - 11q^{59} - q^{61} - 2q^{63} + 4q^{65} + 5q^{67} + 2q^{69} + 13q^{71} - 3q^{75} + 20q^{79} + 2q^{81} + 8q^{83} - 3q^{85} + 2q^{87} + 3q^{89} + 5q^{91} - 2q^{93} + 17q^{95} - 22q^{97} + O(q^{100}) \)

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
−1.30278
2.30278
0 1.00000 0 −1.30278 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 2.30278 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.bk 2
4.b odd 2 1 1932.2.a.e 2
12.b even 2 1 5796.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.e 2 4.b odd 2 1
5796.2.a.k 2 12.b even 2 1
7728.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -3 - T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -13 + T^{2} \)
$13$ \( 3 + 5 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( 3 - 8 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( -51 + 2 T + T^{2} \)
$41$ \( -9 + 4 T + T^{2} \)
$43$ \( -23 - 5 T + T^{2} \)
$47$ \( 36 - 14 T + T^{2} \)
$53$ \( 27 - 15 T + T^{2} \)
$59$ \( 1 + 11 T + T^{2} \)
$61$ \( -81 + T + T^{2} \)
$67$ \( -153 - 5 T + T^{2} \)
$71$ \( 13 - 13 T + T^{2} \)
$73$ \( -117 + T^{2} \)
$79$ \( 87 - 20 T + T^{2} \)
$83$ \( 3 - 8 T + T^{2} \)
$89$ \( -1 - 3 T + T^{2} \)
$97$ \( ( 11 + T )^{2} \)
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