Properties

Label 7938.2.a.bi
Level $7938$
Weight $2$
Character orbit 7938.a
Self dual yes
Analytic conductor $63.385$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta q^{5} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta q^{5} - q^{8} -\beta q^{10} + ( -3 + \beta ) q^{11} + q^{13} + q^{16} + ( 3 - 2 \beta ) q^{17} + ( 1 - 3 \beta ) q^{19} + \beta q^{20} + ( 3 - \beta ) q^{22} + ( -3 - \beta ) q^{23} -2 q^{25} - q^{26} + ( -3 + 2 \beta ) q^{29} + ( 1 + 3 \beta ) q^{31} - q^{32} + ( -3 + 2 \beta ) q^{34} + ( 2 + 3 \beta ) q^{37} + ( -1 + 3 \beta ) q^{38} -\beta q^{40} + ( 6 + 2 \beta ) q^{41} + ( 2 - 6 \beta ) q^{43} + ( -3 + \beta ) q^{44} + ( 3 + \beta ) q^{46} + ( -3 - 3 \beta ) q^{47} + 2 q^{50} + q^{52} + ( -6 + 2 \beta ) q^{53} + ( 3 - 3 \beta ) q^{55} + ( 3 - 2 \beta ) q^{58} + ( -3 + 3 \beta ) q^{59} + ( 1 + 6 \beta ) q^{61} + ( -1 - 3 \beta ) q^{62} + q^{64} + \beta q^{65} + ( -1 - 3 \beta ) q^{67} + ( 3 - 2 \beta ) q^{68} + ( -6 - 6 \beta ) q^{71} + ( 4 - 3 \beta ) q^{73} + ( -2 - 3 \beta ) q^{74} + ( 1 - 3 \beta ) q^{76} + ( -1 + 3 \beta ) q^{79} + \beta q^{80} + ( -6 - 2 \beta ) q^{82} + ( 3 + \beta ) q^{83} + ( -6 + 3 \beta ) q^{85} + ( -2 + 6 \beta ) q^{86} + ( 3 - \beta ) q^{88} + ( -9 + 2 \beta ) q^{89} + ( -3 - \beta ) q^{92} + ( 3 + 3 \beta ) q^{94} + ( -9 + \beta ) q^{95} + 16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{11} + 2q^{13} + 2q^{16} + 6q^{17} + 2q^{19} + 6q^{22} - 6q^{23} - 4q^{25} - 2q^{26} - 6q^{29} + 2q^{31} - 2q^{32} - 6q^{34} + 4q^{37} - 2q^{38} + 12q^{41} + 4q^{43} - 6q^{44} + 6q^{46} - 6q^{47} + 4q^{50} + 2q^{52} - 12q^{53} + 6q^{55} + 6q^{58} - 6q^{59} + 2q^{61} - 2q^{62} + 2q^{64} - 2q^{67} + 6q^{68} - 12q^{71} + 8q^{73} - 4q^{74} + 2q^{76} - 2q^{79} - 12q^{82} + 6q^{83} - 12q^{85} - 4q^{86} + 6q^{88} - 18q^{89} - 6q^{92} + 6q^{94} - 18q^{95} + 32q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
−1.00000 0 1.00000 −1.73205 0 0 −1.00000 0 1.73205
1.2 −1.00000 0 1.00000 1.73205 0 0 −1.00000 0 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)

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 7938.2.a.bi 2
3.b odd 2 1 7938.2.a.br 2
7.b odd 2 1 1134.2.a.j 2
21.c even 2 1 1134.2.a.o yes 2
28.d even 2 1 9072.2.a.bi 2
63.l odd 6 2 1134.2.f.t 4
63.o even 6 2 1134.2.f.q 4
84.h odd 2 1 9072.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.j 2 7.b odd 2 1
1134.2.a.o yes 2 21.c even 2 1
1134.2.f.q 4 63.o even 6 2
1134.2.f.t 4 63.l odd 6 2
7938.2.a.bi 2 1.a even 1 1 trivial
7938.2.a.br 2 3.b odd 2 1
9072.2.a.bf 2 84.h odd 2 1
9072.2.a.bi 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5}^{2} - 3 \)
\( T_{11}^{2} + 6 T_{11} + 6 \)
\( T_{13} - 1 \)
\( T_{17}^{2} - 6 T_{17} - 3 \)
\( T_{23}^{2} + 6 T_{23} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 6 + 6 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -3 - 6 T + T^{2} \)
$19$ \( -26 - 2 T + T^{2} \)
$23$ \( 6 + 6 T + T^{2} \)
$29$ \( -3 + 6 T + T^{2} \)
$31$ \( -26 - 2 T + T^{2} \)
$37$ \( -23 - 4 T + T^{2} \)
$41$ \( 24 - 12 T + T^{2} \)
$43$ \( -104 - 4 T + T^{2} \)
$47$ \( -18 + 6 T + T^{2} \)
$53$ \( 24 + 12 T + T^{2} \)
$59$ \( -18 + 6 T + T^{2} \)
$61$ \( -107 - 2 T + T^{2} \)
$67$ \( -26 + 2 T + T^{2} \)
$71$ \( -72 + 12 T + T^{2} \)
$73$ \( -11 - 8 T + T^{2} \)
$79$ \( -26 + 2 T + T^{2} \)
$83$ \( 6 - 6 T + T^{2} \)
$89$ \( 69 + 18 T + T^{2} \)
$97$ \( ( -16 + T )^{2} \)
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