Properties

Label 7938.2.a.ct
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: \(\Q(\sqrt{2}, \sqrt{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 16x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_1 q^{5} + q^{8} + \beta_1 q^{10} + (\beta_{3} + 1) q^{11} + ( - \beta_{2} - \beta_1) q^{13} + q^{16} + ( - 2 \beta_{2} - \beta_1) q^{17} + ( - 3 \beta_{2} + 2 \beta_1) q^{19} + \beta_1 q^{20} + (\beta_{3} + 1) q^{22} + (\beta_{3} + 5) q^{23} + (\beta_{3} + 3) q^{25} + ( - \beta_{2} - \beta_1) q^{26} + ( - 2 \beta_{3} - 1) q^{29} + 3 \beta_{2} q^{31} + q^{32} + ( - 2 \beta_{2} - \beta_1) q^{34} + q^{37} + ( - 3 \beta_{2} + 2 \beta_1) q^{38} + \beta_1 q^{40} - \beta_{2} q^{41} + 2 q^{43} + (\beta_{3} + 1) q^{44} + (\beta_{3} + 5) q^{46} + (\beta_{2} + 2 \beta_1) q^{47} + (\beta_{3} + 3) q^{50} + ( - \beta_{2} - \beta_1) q^{52} + ( - 2 \beta_{3} + 4) q^{53} + (7 \beta_{2} + 2 \beta_1) q^{55} + ( - 2 \beta_{3} - 1) q^{58} + (3 \beta_{2} - 2 \beta_1) q^{59} + (2 \beta_{2} + 3 \beta_1) q^{61} + 3 \beta_{2} q^{62} + q^{64} + ( - 2 \beta_{3} - 9) q^{65} + (3 \beta_{3} - 3) q^{67} + ( - 2 \beta_{2} - \beta_1) q^{68} + 6 q^{71} + (\beta_{2} - \beta_1) q^{73} + q^{74} + ( - 3 \beta_{2} + 2 \beta_1) q^{76} + ( - \beta_{3} + 5) q^{79} + \beta_1 q^{80} - \beta_{2} q^{82} + 7 \beta_{2} q^{83} + ( - 3 \beta_{3} - 10) q^{85} + 2 q^{86} + (\beta_{3} + 1) q^{88} + (4 \beta_{2} - 3 \beta_1) q^{89} + (\beta_{3} + 5) q^{92} + (\beta_{2} + 2 \beta_1) q^{94} + ( - \beta_{3} + 13) q^{95} + ( - 3 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100}) \) 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 \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 20 q^{23} + 12 q^{25} - 4 q^{29} + 4 q^{32} + 4 q^{37} + 8 q^{43} + 4 q^{44} + 20 q^{46} + 12 q^{50} + 16 q^{53} - 4 q^{58} + 4 q^{64} - 36 q^{65} - 12 q^{67} + 24 q^{71} + 4 q^{74} + 20 q^{79} - 40 q^{85} + 8 q^{86} + 4 q^{88} + 20 q^{92} + 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 16x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 9\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{2} + 9\beta_1 \) 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
−3.44572
−2.03151
2.03151
3.44572
1.00000 0 1.00000 −3.44572 0 0 1.00000 0 −3.44572
1.2 1.00000 0 1.00000 −2.03151 0 0 1.00000 0 −2.03151
1.3 1.00000 0 1.00000 2.03151 0 0 1.00000 0 2.03151
1.4 1.00000 0 1.00000 3.44572 0 0 1.00000 0 3.44572
\(n\): 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.ct yes 4
3.b odd 2 1 7938.2.a.ce 4
7.b odd 2 1 inner 7938.2.a.ct yes 4
21.c even 2 1 7938.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7938.2.a.ce 4 3.b odd 2 1
7938.2.a.ce 4 21.c even 2 1
7938.2.a.ct yes 4 1.a even 1 1 trivial
7938.2.a.ct yes 4 7.b odd 2 1 inner

Hecke kernels

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

\( T_{5}^{4} - 16T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 14 \) Copy content Toggle raw display
\( T_{13}^{4} - 24T_{13}^{2} + 9 \) Copy content Toggle raw display
\( T_{17}^{4} - 40T_{17}^{2} + 25 \) Copy content Toggle raw display
\( T_{23}^{2} - 10T_{23} + 10 \) 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
$5$ \( T^{4} - 16T^{2} + 49 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 24T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} - 40T^{2} + 25 \) Copy content Toggle raw display
$19$ \( T^{4} - 76T^{2} + 484 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10 T + 10)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 59)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 76T^{2} + 484 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 44)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 76T^{2} + 484 \) Copy content Toggle raw display
$61$ \( T^{4} - 184T^{2} + 1849 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 126)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 49 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 10)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 160T^{2} + 3025 \) Copy content Toggle raw display
$97$ \( T^{4} - 76T^{2} + 484 \) Copy content Toggle raw display
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