Properties

Label 7728.2.a.z
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta q^{5} + q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta q^{5} + q^{7} + q^{9} + ( 2 + \beta ) q^{13} + \beta q^{15} + ( -2 + 2 \beta ) q^{19} - q^{21} + q^{23} + ( 5 + \beta ) q^{25} - q^{27} + ( 4 - \beta ) q^{29} -6 q^{31} -\beta q^{35} + \beta q^{37} + ( -2 - \beta ) q^{39} + ( 4 + \beta ) q^{41} + ( -2 - \beta ) q^{43} -\beta q^{45} -\beta q^{47} + q^{49} + ( 2 + 2 \beta ) q^{53} + ( 2 - 2 \beta ) q^{57} + ( 8 - 2 \beta ) q^{59} + 10 q^{61} + q^{63} + ( -10 - 3 \beta ) q^{65} -4 q^{67} - q^{69} -2 \beta q^{71} + ( 2 - 2 \beta ) q^{73} + ( -5 - \beta ) q^{75} + q^{81} + ( 10 - 2 \beta ) q^{83} + ( -4 + \beta ) q^{87} + ( -8 - 2 \beta ) q^{89} + ( 2 + \beta ) q^{91} + 6 q^{93} -20 q^{95} + ( 10 - 3 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - q^{5} + 2q^{7} + 2q^{9} + 5q^{13} + q^{15} - 2q^{19} - 2q^{21} + 2q^{23} + 11q^{25} - 2q^{27} + 7q^{29} - 12q^{31} - q^{35} + q^{37} - 5q^{39} + 9q^{41} - 5q^{43} - q^{45} - q^{47} + 2q^{49} + 6q^{53} + 2q^{57} + 14q^{59} + 20q^{61} + 2q^{63} - 23q^{65} - 8q^{67} - 2q^{69} - 2q^{71} + 2q^{73} - 11q^{75} + 2q^{81} + 18q^{83} - 7q^{87} - 18q^{89} + 5q^{91} + 12q^{93} - 40q^{95} + 17q^{97} + O(q^{100}) \)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -10 + T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -4 - 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -40 + 2 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 2 - 7 T + T^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( -10 - T + T^{2} \)
$41$ \( 10 - 9 T + T^{2} \)
$43$ \( -4 + 5 T + T^{2} \)
$47$ \( -10 + T + T^{2} \)
$53$ \( -32 - 6 T + T^{2} \)
$59$ \( 8 - 14 T + T^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( -40 + 2 T + T^{2} \)
$73$ \( -40 - 2 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 40 - 18 T + T^{2} \)
$89$ \( 40 + 18 T + T^{2} \)
$97$ \( -20 - 17 T + T^{2} \)
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