Properties

Label 7938.2.a.bh
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -1 - \beta ) q^{5} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -1 - \beta ) q^{5} - q^{8} + ( 1 + \beta ) q^{10} + ( 2 - \beta ) q^{11} -2 q^{13} + q^{16} + ( 2 - \beta ) q^{17} -5 q^{19} + ( -1 - \beta ) q^{20} + ( -2 + \beta ) q^{22} + ( -5 + \beta ) q^{23} + ( 4 + 3 \beta ) q^{25} + 2 q^{26} + ( -2 - 2 \beta ) q^{29} -2 q^{31} - q^{32} + ( -2 + \beta ) q^{34} + 2 q^{37} + 5 q^{38} + ( 1 + \beta ) q^{40} + ( -8 + \beta ) q^{41} + ( 2 - 3 \beta ) q^{43} + ( 2 - \beta ) q^{44} + ( 5 - \beta ) q^{46} + ( -4 - 3 \beta ) q^{50} -2 q^{52} + ( -2 - 2 \beta ) q^{53} + 6 q^{55} + ( 2 + 2 \beta ) q^{58} + 3 \beta q^{59} + ( 7 - 3 \beta ) q^{61} + 2 q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} + ( 8 - 3 \beta ) q^{67} + ( 2 - \beta ) q^{68} + ( 3 - 3 \beta ) q^{71} + ( -2 - 3 \beta ) q^{73} -2 q^{74} -5 q^{76} + ( 5 - 3 \beta ) q^{79} + ( -1 - \beta ) q^{80} + ( 8 - \beta ) q^{82} + ( -4 - 4 \beta ) q^{83} + 6 q^{85} + ( -2 + 3 \beta ) q^{86} + ( -2 + \beta ) q^{88} + ( -8 - 2 \beta ) q^{89} + ( -5 + \beta ) q^{92} + ( 5 + 5 \beta ) q^{95} + ( -2 + 3 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 3q^{5} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 3q^{5} - 2q^{8} + 3q^{10} + 3q^{11} - 4q^{13} + 2q^{16} + 3q^{17} - 10q^{19} - 3q^{20} - 3q^{22} - 9q^{23} + 11q^{25} + 4q^{26} - 6q^{29} - 4q^{31} - 2q^{32} - 3q^{34} + 4q^{37} + 10q^{38} + 3q^{40} - 15q^{41} + q^{43} + 3q^{44} + 9q^{46} - 11q^{50} - 4q^{52} - 6q^{53} + 12q^{55} + 6q^{58} + 3q^{59} + 11q^{61} + 4q^{62} + 2q^{64} + 6q^{65} + 13q^{67} + 3q^{68} + 3q^{71} - 7q^{73} - 4q^{74} - 10q^{76} + 7q^{79} - 3q^{80} + 15q^{82} - 12q^{83} + 12q^{85} - q^{86} - 3q^{88} - 18q^{89} - 9q^{92} + 15q^{95} - q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
−1.00000 0 1.00000 −4.37228 0 0 −1.00000 0 4.37228
1.2 −1.00000 0 1.00000 1.37228 0 0 −1.00000 0 −1.37228
\(n\): 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.bh 2
3.b odd 2 1 7938.2.a.bs 2
7.b odd 2 1 1134.2.a.k 2
9.c even 3 2 882.2.f.k 4
9.d odd 6 2 2646.2.f.j 4
21.c even 2 1 1134.2.a.n 2
28.d even 2 1 9072.2.a.bm 2
63.g even 3 2 882.2.h.n 4
63.h even 3 2 882.2.e.k 4
63.i even 6 2 2646.2.e.n 4
63.j odd 6 2 2646.2.e.m 4
63.k odd 6 2 882.2.h.m 4
63.l odd 6 2 126.2.f.d 4
63.n odd 6 2 2646.2.h.l 4
63.o even 6 2 378.2.f.c 4
63.s even 6 2 2646.2.h.k 4
63.t odd 6 2 882.2.e.l 4
84.h odd 2 1 9072.2.a.bb 2
252.s odd 6 2 3024.2.r.f 4
252.bi even 6 2 1008.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 63.l odd 6 2
378.2.f.c 4 63.o even 6 2
882.2.e.k 4 63.h even 3 2
882.2.e.l 4 63.t odd 6 2
882.2.f.k 4 9.c even 3 2
882.2.h.m 4 63.k odd 6 2
882.2.h.n 4 63.g even 3 2
1008.2.r.f 4 252.bi even 6 2
1134.2.a.k 2 7.b odd 2 1
1134.2.a.n 2 21.c even 2 1
2646.2.e.m 4 63.j odd 6 2
2646.2.e.n 4 63.i even 6 2
2646.2.f.j 4 9.d odd 6 2
2646.2.h.k 4 63.s even 6 2
2646.2.h.l 4 63.n odd 6 2
3024.2.r.f 4 252.s odd 6 2
7938.2.a.bh 2 1.a even 1 1 trivial
7938.2.a.bs 2 3.b odd 2 1
9072.2.a.bb 2 84.h odd 2 1
9072.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5}^{2} + 3 T_{5} - 6 \)
\( T_{11}^{2} - 3 T_{11} - 6 \)
\( T_{13} + 2 \)
\( T_{17}^{2} - 3 T_{17} - 6 \)
\( T_{23}^{2} + 9 T_{23} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -6 + 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -6 - 3 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( -6 - 3 T + T^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( 12 + 9 T + T^{2} \)
$29$ \( -24 + 6 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 48 + 15 T + T^{2} \)
$43$ \( -74 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( -24 + 6 T + T^{2} \)
$59$ \( -72 - 3 T + T^{2} \)
$61$ \( -44 - 11 T + T^{2} \)
$67$ \( -32 - 13 T + T^{2} \)
$71$ \( -72 - 3 T + T^{2} \)
$73$ \( -62 + 7 T + T^{2} \)
$79$ \( -62 - 7 T + T^{2} \)
$83$ \( -96 + 12 T + T^{2} \)
$89$ \( 48 + 18 T + T^{2} \)
$97$ \( -74 + T + T^{2} \)
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