Properties

Label 7728.2.a.bc
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} - q^{7} + q^{9} + 4 q^{11} + ( 2 + \beta ) q^{13} -\beta q^{15} + 4 q^{17} + ( -2 + 2 \beta ) q^{19} + q^{21} - q^{23} + ( 5 + \beta ) q^{25} - q^{27} + ( 4 - \beta ) q^{29} + 2 q^{31} -4 q^{33} -\beta q^{35} + ( 8 - \beta ) q^{37} + ( -2 - \beta ) q^{39} + ( -4 + \beta ) q^{41} + ( -2 + \beta ) q^{43} + \beta q^{45} + 3 \beta q^{47} + q^{49} -4 q^{51} + ( -2 - 2 \beta ) q^{53} + 4 \beta q^{55} + ( 2 - 2 \beta ) q^{57} + 2 \beta q^{59} -2 q^{61} - q^{63} + ( 10 + 3 \beta ) q^{65} -4 \beta q^{67} + q^{69} + 2 \beta q^{71} + ( -6 + 2 \beta ) q^{73} + ( -5 - \beta ) q^{75} -4 q^{77} -8 q^{79} + q^{81} + ( -6 - 2 \beta ) q^{83} + 4 \beta q^{85} + ( -4 + \beta ) q^{87} + ( -4 - 2 \beta ) q^{89} + ( -2 - \beta ) q^{91} -2 q^{93} + 20 q^{95} + ( 10 - \beta ) q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + 8q^{11} + 5q^{13} - q^{15} + 8q^{17} - 2q^{19} + 2q^{21} - 2q^{23} + 11q^{25} - 2q^{27} + 7q^{29} + 4q^{31} - 8q^{33} - q^{35} + 15q^{37} - 5q^{39} - 7q^{41} - 3q^{43} + q^{45} + 3q^{47} + 2q^{49} - 8q^{51} - 6q^{53} + 4q^{55} + 2q^{57} + 2q^{59} - 4q^{61} - 2q^{63} + 23q^{65} - 4q^{67} + 2q^{69} + 2q^{71} - 10q^{73} - 11q^{75} - 8q^{77} - 16q^{79} + 2q^{81} - 14q^{83} + 4q^{85} - 7q^{87} - 10q^{89} - 5q^{91} - 4q^{93} + 40q^{95} + 19q^{97} + 8q^{99} + O(q^{100}) \)

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
−2.70156
3.70156
0 −1.00000 0 −2.70156 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 3.70156 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.bc 2
4.b odd 2 1 966.2.a.n 2
12.b even 2 1 2898.2.a.bb 2
28.d even 2 1 6762.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.n 2 4.b odd 2 1
2898.2.a.bb 2 12.b even 2 1
6762.2.a.bo 2 28.d even 2 1
7728.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{2} - T_{5} - 10 \)
\( T_{11} - 4 \)
\( T_{13}^{2} - 5 T_{13} - 4 \)
\( T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -10 - T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -4 - 5 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( -40 + 2 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( 2 - 7 T + T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 46 - 15 T + T^{2} \)
$41$ \( 2 + 7 T + T^{2} \)
$43$ \( -8 + 3 T + T^{2} \)
$47$ \( -90 - 3 T + T^{2} \)
$53$ \( -32 + 6 T + T^{2} \)
$59$ \( -40 - 2 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -160 + 4 T + T^{2} \)
$71$ \( -40 - 2 T + T^{2} \)
$73$ \( -16 + 10 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 8 + 14 T + T^{2} \)
$89$ \( -16 + 10 T + T^{2} \)
$97$ \( 80 - 19 T + T^{2} \)
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