Properties

Label 7728.2.a.bi
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: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{7} + q^{9} + \beta q^{11} + 2 q^{13} + ( -2 + 2 \beta ) q^{17} + ( 4 + \beta ) q^{19} - q^{21} - q^{23} -5 q^{25} + q^{27} + 2 q^{29} + ( 4 - 2 \beta ) q^{31} + \beta q^{33} + ( -4 + 2 \beta ) q^{37} + 2 q^{39} -\beta q^{41} + ( -6 + 2 \beta ) q^{43} + ( 10 - \beta ) q^{47} + q^{49} + ( -2 + 2 \beta ) q^{51} + ( 2 - 3 \beta ) q^{53} + ( 4 + \beta ) q^{57} + ( 6 - 3 \beta ) q^{59} + ( 10 - \beta ) q^{61} - q^{63} + ( 6 + 2 \beta ) q^{67} - q^{69} + ( 2 - 2 \beta ) q^{73} -5 q^{75} -\beta q^{77} + ( -4 - 2 \beta ) q^{79} + q^{81} + 10 q^{83} + 2 q^{87} -2 q^{89} -2 q^{91} + ( 4 - 2 \beta ) q^{93} + 6 q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{7} + 2q^{9} + q^{11} + 4q^{13} - 2q^{17} + 9q^{19} - 2q^{21} - 2q^{23} - 10q^{25} + 2q^{27} + 4q^{29} + 6q^{31} + q^{33} - 6q^{37} + 4q^{39} - q^{41} - 10q^{43} + 19q^{47} + 2q^{49} - 2q^{51} + q^{53} + 9q^{57} + 9q^{59} + 19q^{61} - 2q^{63} + 14q^{67} - 2q^{69} + 2q^{73} - 10q^{75} - q^{77} - 10q^{79} + 2q^{81} + 20q^{83} + 4q^{87} - 4q^{89} - 4q^{91} + 6q^{93} + 12q^{97} + q^{99} + O(q^{100}) \)

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

\( T_{5} \)
\( T_{11}^{2} - T_{11} - 10 \)
\( T_{13} - 2 \)
\( T_{17}^{2} + 2 T_{17} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -10 - T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -40 + 2 T + T^{2} \)
$19$ \( 10 - 9 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( -32 - 6 T + T^{2} \)
$37$ \( -32 + 6 T + T^{2} \)
$41$ \( -10 + T + T^{2} \)
$43$ \( -16 + 10 T + T^{2} \)
$47$ \( 80 - 19 T + T^{2} \)
$53$ \( -92 - T + T^{2} \)
$59$ \( -72 - 9 T + T^{2} \)
$61$ \( 80 - 19 T + T^{2} \)
$67$ \( 8 - 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -40 - 2 T + T^{2} \)
$79$ \( -16 + 10 T + T^{2} \)
$83$ \( ( -10 + T )^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( ( -6 + T )^{2} \)
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