Properties

Label 5148.2.a.l
Level $5148$
Weight $2$
Character orbit 5148.a
Self dual yes
Analytic conductor $41.107$
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] = [5148,2,Mod(1,5148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("5148.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5148.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: \(41.1069869606\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1716)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 2 \beta_1 - 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 2 \beta_1 - 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7} + q^{11} + q^{13} + (\beta_1 - 1) q^{17} + ( - 3 \beta_{2} + \beta_1 - 2) q^{19} + ( - \beta_{2} + \beta_1 - 4) q^{23} + (\beta_{2} + \beta_1 + 9) q^{25} + ( - 2 \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_{2} - 2 \beta_1 + 7) q^{31} - 6 \beta_{2} q^{35} + ( - 2 \beta_{2} + 2) q^{37} + (\beta_{2} + \beta_1 - 4) q^{41} + ( - 2 \beta_{2} + \beta_1 - 5) q^{43} + (2 \beta_{2} + 2 \beta_1 - 2) q^{47} + ( - 4 \beta_{2} + 2 \beta_1 + 7) q^{49} + (2 \beta_{2} + 4 \beta_1 - 4) q^{53} + (\beta_{2} - 2 \beta_1 - 1) q^{55} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{59} - 4 \beta_1 q^{61} + (\beta_{2} - 2 \beta_1 - 1) q^{65} + (\beta_{2} + 4 \beta_1 - 5) q^{67} - 2 \beta_{2} q^{71} + (3 \beta_{2} + \beta_1 + 4) q^{73} + (\beta_{2} + \beta_1 - 2) q^{77} + (4 \beta_{2} - 7 \beta_1 - 3) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{83} + ( - 3 \beta_{2} + \beta_1 - 4) q^{85} + ( - 5 \beta_{2} - 2 \beta_1 - 1) q^{89} + (\beta_{2} + \beta_1 - 2) q^{91} + (2 \beta_{2} + 12 \beta_1 - 12) q^{95} + (2 \beta_{2} - 4 \beta_1 - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{5} - 4 q^{7} + 3 q^{11} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 12 q^{23} + 29 q^{25} - 4 q^{29} + 18 q^{31} - 6 q^{35} + 4 q^{37} - 10 q^{41} - 16 q^{43} - 2 q^{47} + 19 q^{49} - 6 q^{53} - 4 q^{55} + 8 q^{59} - 4 q^{61} - 4 q^{65} - 10 q^{67} - 2 q^{71} + 16 q^{73} - 4 q^{77} - 12 q^{79} + 8 q^{83} - 14 q^{85} - 10 q^{89} - 4 q^{91} - 22 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x - 1 \) : 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
2.86620
−0.210756
−1.65544
0 0 0 −4.38350 0 3.21509 0 0 0
1.2 0 0 0 −3.32331 0 −4.95558 0 0 0
1.3 0 0 0 3.70682 0 −2.25951 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 5148.2.a.l 3
3.b odd 2 1 1716.2.a.h 3
12.b even 2 1 6864.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.h 3 3.b odd 2 1
5148.2.a.l 3 1.a even 1 1 trivial
6864.2.a.bt 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5148))\):

\( T_{5}^{3} + 4T_{5}^{2} - 14T_{5} - 54 \) Copy content Toggle raw display
\( T_{7}^{3} + 4T_{7}^{2} - 12T_{7} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 14 T - 54 \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 12 T - 36 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 4 T - 6 \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} - 36 T - 292 \) Copy content Toggle raw display
$23$ \( T^{3} + 12 T^{2} + 40 T + 36 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 20 T - 66 \) Copy content Toggle raw display
$31$ \( T^{3} - 18 T^{2} + 70 T + 98 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} - 24 T - 16 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + 16 T - 36 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} + 60 T - 18 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 68 T - 168 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 140 T - 984 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 56 T + 432 \) Copy content Toggle raw display
$61$ \( T^{3} + 4 T^{2} - 80 T + 64 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} - 78 T - 774 \) Copy content Toggle raw display
$71$ \( T^{3} + 2 T^{2} - 28 T - 72 \) Copy content Toggle raw display
$73$ \( T^{3} - 16T^{2} + 404 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} - 200 T - 2422 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} - 56 T + 432 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 218 T - 1134 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} + 56 T - 464 \) Copy content Toggle raw display
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