Properties

Label 7938.2.a.bx
Level $7938$
Weight $2$
Character orbit 7938.a
Self dual yes
Analytic conductor $63.385$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7938,2,Mod(1,7938)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7938, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7938.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} - q^{8} + (\beta_1 - 2) q^{10} + (2 \beta_{2} + \beta_1) q^{11} + (\beta_1 - 1) q^{13} + q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_1 + 2) q^{20} + ( - 2 \beta_{2} - \beta_1) q^{22} + ( - \beta_{2} + \beta_1 - 3) q^{23} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + ( - \beta_1 + 1) q^{26} + ( - \beta_{2} - 3 \beta_1 - 1) q^{29} + (\beta_{2} + 2 \beta_1 - 5) q^{31} - q^{32} + (2 \beta_{2} - 2 \beta_1) q^{34} + (2 \beta_{2} + 2 \beta_1 + 3) q^{37} + (\beta_{2} + \beta_1 + 1) q^{38} + (\beta_1 - 2) q^{40} + ( - \beta_{2} - \beta_1 + 4) q^{41} + (\beta_{2} - 2 \beta_1 - 5) q^{43} + (2 \beta_{2} + \beta_1) q^{44} + (\beta_{2} - \beta_1 + 3) q^{46} + ( - \beta_{2} + 2 \beta_1 - 2) q^{47} + ( - \beta_{2} + 3 \beta_1 - 2) q^{50} + (\beta_1 - 1) q^{52} + (\beta_{2} + 4 \beta_1 + 2) q^{53} + (3 \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{2} + 3 \beta_1 + 1) q^{58} + ( - \beta_{2} - 5 \beta_1) q^{59} + ( - 3 \beta_{2} - 5 \beta_1 + 2) q^{61} + ( - \beta_{2} - 2 \beta_1 + 5) q^{62} + q^{64} + ( - \beta_{2} + 2 \beta_1 - 5) q^{65} + ( - 3 \beta_{2} + \beta_1 - 3) q^{67} + ( - 2 \beta_{2} + 2 \beta_1) q^{68} + (3 \beta_{2} - \beta_1 - 1) q^{71} + ( - 3 \beta_{2} - \beta_1 - 9) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{74} + ( - \beta_{2} - \beta_1 - 1) q^{76} + (4 \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_1 + 2) q^{80} + (\beta_{2} + \beta_1 - 4) q^{82} + ( - 2 \beta_{2} - \beta_1 - 3) q^{83} + ( - 6 \beta_{2} + 4 \beta_1 - 8) q^{85} + ( - \beta_{2} + 2 \beta_1 + 5) q^{86} + ( - 2 \beta_{2} - \beta_1) q^{88} + (2 \beta_{2} - \beta_1 + 4) q^{89} + ( - \beta_{2} + \beta_1 - 3) q^{92} + (\beta_{2} - 2 \beta_1 + 2) q^{94} + ( - \beta_{2} + \beta_1) q^{95} + ( - 2 \beta_{2} - 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{8} - 5 q^{10} - q^{11} - 2 q^{13} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} + 2 q^{26} - 5 q^{29} - 14 q^{31} - 3 q^{32} - 4 q^{34} + 9 q^{37} + 3 q^{38} - 5 q^{40} + 12 q^{41} - 18 q^{43} - q^{44} + 7 q^{46} - 3 q^{47} - 2 q^{50} - 2 q^{52} + 9 q^{53} - 7 q^{55} + 5 q^{58} - 4 q^{59} + 4 q^{61} + 14 q^{62} + 3 q^{64} - 12 q^{65} - 5 q^{67} + 4 q^{68} - 7 q^{71} - 25 q^{73} - 9 q^{74} - 3 q^{76} - 7 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} - 14 q^{85} + 18 q^{86} + q^{88} + 9 q^{89} - 7 q^{92} + 3 q^{94} + 2 q^{95} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.46050
0.239123
−1.69963
−1.00000 0 1.00000 −0.460505 0 0 −1.00000 0 0.460505
1.2 −1.00000 0 1.00000 1.76088 0 0 −1.00000 0 −1.76088
1.3 −1.00000 0 1.00000 3.69963 0 0 −1.00000 0 −3.69963
\(n\): 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.bx 3
3.b odd 2 1 7938.2.a.by 3
7.b odd 2 1 7938.2.a.bu 3
7.d odd 6 2 1134.2.g.n 6
9.c even 3 2 2646.2.f.n 6
9.d odd 6 2 882.2.f.m 6
21.c even 2 1 7938.2.a.cb 3
21.g even 6 2 1134.2.g.k 6
63.g even 3 2 2646.2.h.p 6
63.h even 3 2 2646.2.e.o 6
63.i even 6 2 126.2.e.d 6
63.j odd 6 2 882.2.e.p 6
63.k odd 6 2 378.2.h.d 6
63.l odd 6 2 2646.2.f.o 6
63.n odd 6 2 882.2.h.o 6
63.o even 6 2 882.2.f.l 6
63.s even 6 2 126.2.h.c yes 6
63.t odd 6 2 378.2.e.c 6
252.n even 6 2 3024.2.t.g 6
252.r odd 6 2 1008.2.q.h 6
252.bj even 6 2 3024.2.q.h 6
252.bn odd 6 2 1008.2.t.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 63.i even 6 2
126.2.h.c yes 6 63.s even 6 2
378.2.e.c 6 63.t odd 6 2
378.2.h.d 6 63.k odd 6 2
882.2.e.p 6 63.j odd 6 2
882.2.f.l 6 63.o even 6 2
882.2.f.m 6 9.d odd 6 2
882.2.h.o 6 63.n odd 6 2
1008.2.q.h 6 252.r odd 6 2
1008.2.t.g 6 252.bn odd 6 2
1134.2.g.k 6 21.g even 6 2
1134.2.g.n 6 7.d odd 6 2
2646.2.e.o 6 63.h even 3 2
2646.2.f.n 6 9.c even 3 2
2646.2.f.o 6 63.l odd 6 2
2646.2.h.p 6 63.g even 3 2
3024.2.q.h 6 252.bj even 6 2
3024.2.t.g 6 252.n even 6 2
7938.2.a.bu 3 7.b odd 2 1
7938.2.a.bx 3 1.a even 1 1 trivial
7938.2.a.by 3 3.b odd 2 1
7938.2.a.cb 3 21.c even 2 1

Hecke kernels

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

\( T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 26T_{11} + 33 \) Copy content Toggle raw display
\( T_{13}^{3} + 2T_{13}^{2} - 3T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{3} - 4T_{17}^{2} - 44T_{17} + 168 \) Copy content Toggle raw display
\( T_{23}^{3} + 7T_{23}^{2} + 4T_{23} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 5 T^{2} + 4 T + 3 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 26 T + 33 \) Copy content Toggle raw display
$13$ \( T^{3} + 2 T^{2} - 3 T - 3 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 44 T + 168 \) Copy content Toggle raw display
$19$ \( T^{3} + 3 T^{2} - 6 T - 7 \) Copy content Toggle raw display
$23$ \( T^{3} + 7 T^{2} + 4 T - 3 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} - 32 T + 33 \) Copy content Toggle raw display
$31$ \( T^{3} + 14 T^{2} + 45 T - 27 \) Copy content Toggle raw display
$37$ \( T^{3} - 9 T^{2} - 9 T + 73 \) Copy content Toggle raw display
$41$ \( T^{3} - 12 T^{2} + 39 T - 27 \) Copy content Toggle raw display
$43$ \( T^{3} + 18 T^{2} + 81 T + 1 \) Copy content Toggle raw display
$47$ \( T^{3} + 3 T^{2} - 24 T + 27 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} - 42 T - 9 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} - 101 T + 177 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} - 135 T + 717 \) Copy content Toggle raw display
$67$ \( T^{3} + 5 T^{2} - 58 T - 149 \) Copy content Toggle raw display
$71$ \( T^{3} + 7 T^{2} - 50 T - 99 \) Copy content Toggle raw display
$73$ \( T^{3} + 25 T^{2} + 152 T - 49 \) Copy content Toggle raw display
$79$ \( T^{3} + 7 T^{2} - 144 T - 771 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} - 5 T - 93 \) Copy content Toggle raw display
$89$ \( T^{3} - 9 T^{2} - 6 T + 63 \) Copy content Toggle raw display
$97$ \( T^{3} + 28 T^{2} + 236 T + 536 \) Copy content Toggle raw display
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