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Properties

Label 7938.2.a.ch
Level $7938$
Weight $2$
Character orbit 7938.a
Self dual yes
Analytic conductor $63.385$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{7}, \sqrt{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 11x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 882)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta_{3} q^{5} - q^{8} - \beta_{3} q^{10} - \beta_{2} q^{11} - 2 \beta_1 q^{13} + q^{16} + (2 \beta_{3} + \beta_1) q^{17} + ( - \beta_{3} - \beta_1) q^{19} + \beta_{3} q^{20} + \beta_{2} q^{22} + ( - \beta_{2} + 1) q^{23} - \beta_{2} q^{25} + 2 \beta_1 q^{26} - 4 q^{29} + 2 \beta_1 q^{31} - q^{32} + ( - 2 \beta_{3} - \beta_1) q^{34} + 6 q^{37} + (\beta_{3} + \beta_1) q^{38} - \beta_{3} q^{40} + (2 \beta_{3} + \beta_1) q^{41} + ( - \beta_{2} + 2) q^{43} - \beta_{2} q^{44} + (\beta_{2} - 1) q^{46} + ( - 4 \beta_{3} - 4 \beta_1) q^{47} + \beta_{2} q^{50} - 2 \beta_1 q^{52} - 4 q^{53} + (6 \beta_{3} + 2 \beta_1) q^{55} + 4 q^{58} + (4 \beta_{3} - 3 \beta_1) q^{59} + (\beta_{3} - 2 \beta_1) q^{61} - 2 \beta_1 q^{62} + q^{64} + 4 q^{65} + \beta_{2} q^{67} + (2 \beta_{3} + \beta_1) q^{68} + (\beta_{2} - 5) q^{71} + ( - 4 \beta_{3} - 3 \beta_1) q^{73} - 6 q^{74} + ( - \beta_{3} - \beta_1) q^{76} + (\beta_{2} + 9) q^{79} + \beta_{3} q^{80} + ( - 2 \beta_{3} - \beta_1) q^{82} + (2 \beta_{3} - 2 \beta_1) q^{83} + ( - 2 \beta_{2} + 8) q^{85} + (\beta_{2} - 2) q^{86} + \beta_{2} q^{88} + ( - 2 \beta_{3} + 4 \beta_1) q^{89} + ( - \beta_{2} + 1) q^{92} + (4 \beta_{3} + 4 \beta_1) q^{94} + (\beta_{2} - 3) q^{95} + 5 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{11} + 4 q^{16} - 2 q^{22} + 6 q^{23} + 2 q^{25} - 16 q^{29} - 4 q^{32} + 24 q^{37} + 10 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{50} - 16 q^{53} + 16 q^{58} + 4 q^{64} + 16 q^{65} - 2 q^{67} - 22 q^{71} - 24 q^{74} + 34 q^{79} + 36 q^{85} - 10 q^{86} - 2 q^{88} + 6 q^{92} - 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 11x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 11\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 11\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.613616
3.25937
−3.25937
−0.613616
−1.00000 0 1.00000 −3.25937 0 0 −1.00000 0 3.25937
1.2 −1.00000 0 1.00000 −0.613616 0 0 −1.00000 0 0.613616
1.3 −1.00000 0 1.00000 0.613616 0 0 −1.00000 0 −0.613616
1.4 −1.00000 0 1.00000 3.25937 0 0 −1.00000 0 −3.25937
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7938.2.a.ch 4
3.b odd 2 1 7938.2.a.cq 4
7.b odd 2 1 inner 7938.2.a.ch 4
9.c even 3 2 882.2.f.r ✓ 8
9.d odd 6 2 2646.2.f.p 8
21.c even 2 1 7938.2.a.cq 4
63.g even 3 2 882.2.h.s 8
63.h even 3 2 882.2.e.r 8
63.i even 6 2 2646.2.e.s 8
63.j odd 6 2 2646.2.e.s 8
63.k odd 6 2 882.2.h.s 8
63.l odd 6 2 882.2.f.r ✓ 8
63.n odd 6 2 2646.2.h.r 8
63.o even 6 2 2646.2.f.p 8
63.s even 6 2 2646.2.h.r 8
63.t odd 6 2 882.2.e.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.r 8 63.h even 3 2
882.2.e.r 8 63.t odd 6 2
882.2.f.r ✓ 8 9.c even 3 2
882.2.f.r ✓ 8 63.l odd 6 2
882.2.h.s 8 63.g even 3 2
882.2.h.s 8 63.k odd 6 2
2646.2.e.s 8 63.i even 6 2
2646.2.e.s 8 63.j odd 6 2
2646.2.f.p 8 9.d odd 6 2
2646.2.f.p 8 63.o even 6 2
2646.2.h.r 8 63.n odd 6 2
2646.2.h.r 8 63.s even 6 2
7938.2.a.ch 4 1.a even 1 1 trivial
7938.2.a.ch 4 7.b odd 2 1 inner
7938.2.a.cq 4 3.b odd 2 1
7938.2.a.cq 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5}^{4} - 11T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 26 \) Copy content Toggle raw display
\( T_{13}^{4} - 44T_{13}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{4} - 39T_{17}^{2} + 144 \) Copy content Toggle raw display
\( T_{23}^{2} - 3T_{23} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 11T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T - 26)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 44T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} - 39T^{2} + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T - 24)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 44T^{2} + 64 \) Copy content Toggle raw display
$37$ \( (T - 6)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 39T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T^{2} - 5 T - 20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$53$ \( (T + 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 371 T^{2} + 33124 \) Copy content Toggle raw display
$61$ \( T^{4} - 71T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 26)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 11 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 179T^{2} + 6724 \) Copy content Toggle raw display
$79$ \( (T^{2} - 17 T + 46)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 284 T^{2} + 16384 \) Copy content Toggle raw display
$97$ \( T^{4} - 275T^{2} + 2500 \) Copy content Toggle raw display
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