Properties

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

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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: \(1\)
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 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.

\(f(q)\) \(=\) \( q + q^{3} + ( -2 - \beta ) q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -2 - \beta ) q^{5} + q^{7} + q^{9} + ( -3 + 2 \beta ) q^{11} + ( 1 - \beta ) q^{13} + ( -2 - \beta ) q^{15} + ( -3 - 2 \beta ) q^{17} + ( 3 + 2 \beta ) q^{19} + q^{21} + q^{23} + ( 2 + 5 \beta ) q^{25} + q^{27} + ( -3 + 2 \beta ) q^{29} + ( 1 - 2 \beta ) q^{31} + ( -3 + 2 \beta ) q^{33} + ( -2 - \beta ) q^{35} + ( 3 - 2 \beta ) q^{37} + ( 1 - \beta ) q^{39} + ( 1 - 2 \beta ) q^{41} + ( -3 + \beta ) q^{43} + ( -2 - \beta ) q^{45} + ( 2 - 2 \beta ) q^{47} + q^{49} + ( -3 - 2 \beta ) q^{51} + ( 2 - 3 \beta ) q^{53} -3 \beta q^{55} + ( 3 + 2 \beta ) q^{57} + 3 \beta q^{59} + ( 1 - 3 \beta ) q^{61} + q^{63} + ( 1 + 2 \beta ) q^{65} + ( -12 + \beta ) q^{67} + q^{69} + ( -9 + 3 \beta ) q^{71} + 11 q^{73} + ( 2 + 5 \beta ) q^{75} + ( -3 + 2 \beta ) q^{77} + ( -3 + 8 \beta ) q^{79} + q^{81} + ( -11 + 4 \beta ) q^{83} + ( 12 + 9 \beta ) q^{85} + ( -3 + 2 \beta ) q^{87} + ( -9 + 5 \beta ) q^{89} + ( 1 - \beta ) q^{91} + ( 1 - 2 \beta ) q^{93} + ( -12 - 9 \beta ) q^{95} + ( 9 + 4 \beta ) q^{97} + ( -3 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + q^{13} - 5q^{15} - 8q^{17} + 8q^{19} + 2q^{21} + 2q^{23} + 9q^{25} + 2q^{27} - 4q^{29} - 4q^{33} - 5q^{35} + 4q^{37} + q^{39} - 5q^{43} - 5q^{45} + 2q^{47} + 2q^{49} - 8q^{51} + q^{53} - 3q^{55} + 8q^{57} + 3q^{59} - q^{61} + 2q^{63} + 4q^{65} - 23q^{67} + 2q^{69} - 15q^{71} + 22q^{73} + 9q^{75} - 4q^{77} + 2q^{79} + 2q^{81} - 18q^{83} + 33q^{85} - 4q^{87} - 13q^{89} + q^{91} - 33q^{95} + 22q^{97} - 4q^{99} + O(q^{100}) \)

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.bf 2
4.b odd 2 1 1932.2.a.c 2
12.b even 2 1 5796.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.c 2 4.b odd 2 1
5796.2.a.m 2 12.b even 2 1
7728.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} + 5 T_{5} + 3 \)
\( T_{11}^{2} + 4 T_{11} - 9 \)
\( T_{13}^{2} - T_{13} - 3 \)
\( T_{17}^{2} + 8 T_{17} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 3 + 5 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -9 + 4 T + T^{2} \)
$13$ \( -3 - T + T^{2} \)
$17$ \( 3 + 8 T + T^{2} \)
$19$ \( 3 - 8 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -9 + 4 T + T^{2} \)
$31$ \( -13 + T^{2} \)
$37$ \( -9 - 4 T + T^{2} \)
$41$ \( -13 + T^{2} \)
$43$ \( 3 + 5 T + T^{2} \)
$47$ \( -12 - 2 T + T^{2} \)
$53$ \( -29 - T + T^{2} \)
$59$ \( -27 - 3 T + T^{2} \)
$61$ \( -29 + T + T^{2} \)
$67$ \( 129 + 23 T + T^{2} \)
$71$ \( 27 + 15 T + T^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( -207 - 2 T + T^{2} \)
$83$ \( 29 + 18 T + T^{2} \)
$89$ \( -39 + 13 T + T^{2} \)
$97$ \( 69 - 22 T + T^{2} \)
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