Properties

Label 7938.2.a.bq
Level $7938$
Weight $2$
Character orbit 7938.a
Self dual yes
Analytic conductor $63.385$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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 \(\beta = 3\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{8} + \beta q^{11} + ( 2 + \beta ) q^{13} + q^{16} + ( 2 + \beta ) q^{19} + \beta q^{22} + ( 3 + \beta ) q^{23} -5 q^{25} + ( 2 + \beta ) q^{26} -\beta q^{29} + ( 5 - \beta ) q^{31} + q^{32} -4 q^{37} + ( 2 + \beta ) q^{38} + ( 3 - 2 \beta ) q^{41} + ( 2 - 2 \beta ) q^{43} + \beta q^{44} + ( 3 + \beta ) q^{46} + ( 9 + \beta ) q^{47} -5 q^{50} + ( 2 + \beta ) q^{52} + \beta q^{53} -\beta q^{58} + ( 2 + \beta ) q^{61} + ( 5 - \beta ) q^{62} + q^{64} + ( -4 - \beta ) q^{67} + ( -3 - \beta ) q^{71} -7 q^{73} -4 q^{74} + ( 2 + \beta ) q^{76} + ( 5 + \beta ) q^{79} + ( 3 - 2 \beta ) q^{82} + ( 12 - \beta ) q^{83} + ( 2 - 2 \beta ) q^{86} + \beta q^{88} + ( 3 - 2 \beta ) q^{89} + ( 3 + \beta ) q^{92} + ( 9 + \beta ) q^{94} + ( -4 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 4q^{13} + 2q^{16} + 4q^{19} + 6q^{23} - 10q^{25} + 4q^{26} + 10q^{31} + 2q^{32} - 8q^{37} + 4q^{38} + 6q^{41} + 4q^{43} + 6q^{46} + 18q^{47} - 10q^{50} + 4q^{52} + 4q^{61} + 10q^{62} + 2q^{64} - 8q^{67} - 6q^{71} - 14q^{73} - 8q^{74} + 4q^{76} + 10q^{79} + 6q^{82} + 24q^{83} + 4q^{86} + 6q^{89} + 6q^{92} + 18q^{94} - 8q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
1.00000 0 1.00000 0 0 0 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 0 1.00000 0 0
\(n\): 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.bq 2
3.b odd 2 1 7938.2.a.bk 2
7.b odd 2 1 7938.2.a.bp 2
7.c even 3 2 1134.2.g.i 4
21.c even 2 1 7938.2.a.bj 2
21.h odd 6 2 1134.2.g.j yes 4
63.g even 3 2 1134.2.h.r 4
63.h even 3 2 1134.2.e.s 4
63.j odd 6 2 1134.2.e.r 4
63.n odd 6 2 1134.2.h.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.e.r 4 63.j odd 6 2
1134.2.e.s 4 63.h even 3 2
1134.2.g.i 4 7.c even 3 2
1134.2.g.j yes 4 21.h odd 6 2
1134.2.h.r 4 63.g even 3 2
1134.2.h.s 4 63.n odd 6 2
7938.2.a.bj 2 21.c even 2 1
7938.2.a.bk 2 3.b odd 2 1
7938.2.a.bp 2 7.b odd 2 1
7938.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5} \)
\( T_{11}^{2} - 18 \)
\( T_{13}^{2} - 4 T_{13} - 14 \)
\( T_{17} \)
\( T_{23}^{2} - 6 T_{23} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -18 + T^{2} \)
$13$ \( -14 - 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -14 - 4 T + T^{2} \)
$23$ \( -9 - 6 T + T^{2} \)
$29$ \( -18 + T^{2} \)
$31$ \( 7 - 10 T + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( -63 - 6 T + T^{2} \)
$43$ \( -68 - 4 T + T^{2} \)
$47$ \( 63 - 18 T + T^{2} \)
$53$ \( -18 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( -14 - 4 T + T^{2} \)
$67$ \( -2 + 8 T + T^{2} \)
$71$ \( -9 + 6 T + T^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( 7 - 10 T + T^{2} \)
$83$ \( 126 - 24 T + T^{2} \)
$89$ \( -63 - 6 T + T^{2} \)
$97$ \( -56 + 8 T + T^{2} \)
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