Properties

Label 5148.2.a.q
Level $5148$
Weight $2$
Character orbit 5148.a
Self dual yes
Analytic conductor $41.107$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.90996.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 11x^{2} - 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{2} + 1) q^{7} - q^{11} + q^{13} + ( - \beta_{3} + \beta_1) q^{17} + (\beta_{2} + 1) q^{19} + (\beta_{3} + 2 \beta_1 + 2) q^{23} + ( - \beta_{3} + 1) q^{25} + (\beta_{2} - \beta_1 - 3) q^{29} + (\beta_1 + 2) q^{31} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{35} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{37} + (2 \beta_{3} + \beta_{2} - 1) q^{41} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{43} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 - 3) q^{47} + (2 \beta_{3} + 2 \beta_1 + 9) q^{49} + (2 \beta_1 + 2) q^{53} - \beta_1 q^{55} - 2 \beta_1 q^{59} + ( - \beta_{3} - \beta_{2} + 5) q^{61} + \beta_1 q^{65} + (2 \beta_{3} + \beta_1 + 2) q^{67} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 5) q^{71} + (\beta_{2} - 2 \beta_1 + 1) q^{73} + ( - \beta_{2} - 1) q^{77} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 4) q^{79} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{83} + (\beta_{2} + 4 \beta_1 + 11) q^{85} + (\beta_{3} + \beta_{2} - \beta_1 - 11) q^{89} + (\beta_{2} + 1) q^{91} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{95} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 4 q^{25} - 12 q^{29} + 6 q^{31} + 10 q^{35} - 2 q^{37} - 6 q^{41} + 2 q^{43} - 6 q^{47} + 32 q^{49} + 4 q^{53} + 2 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} + 6 q^{67} - 14 q^{71} + 6 q^{73} - 2 q^{77} + 14 q^{79} + 34 q^{85} - 44 q^{89} + 2 q^{91} + 10 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 11x^{2} - 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 9\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 10\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 12\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} - 4\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 15\beta_{2} - 4\beta _1 + 27 ) / 2 \) 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
−2.63415
3.94698
0.309233
−0.622070
0 0 0 −3.22388 0 −0.874887 0 0 0
1.2 0 0 0 −1.59571 0 4.44027 0 0 0
1.3 0 0 0 −0.472388 0 −5.15838 0 0 0
1.4 0 0 0 3.29198 0 3.59300 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.q 4
3.b odd 2 1 1716.2.a.i 4
12.b even 2 1 6864.2.a.cb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.i 4 3.b odd 2 1
5148.2.a.q 4 1.a even 1 1 trivial
6864.2.a.cb 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5148))\):

\( T_{5}^{4} + 2T_{5}^{3} - 10T_{5}^{2} - 22T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 28T_{7}^{2} + 60T_{7} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 10 T^{2} - 22 T - 8 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 28 T^{2} + 60 T + 72 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 64 T^{2} - 186 T + 300 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 28 T^{2} + 60 T + 72 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 64 T^{2} + 380 T - 512 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} + 24 T^{2} + \cdots - 108 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + 2 T^{2} + 10 T - 4 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 88 T^{2} - 112 T - 32 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} - 136 T^{2} + \cdots + 3872 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} - 104 T^{2} + \cdots - 1048 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 188 T^{2} + \cdots + 4768 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} - 40 T^{2} + 48 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} - 40 T^{2} + 176 T - 128 \) Copy content Toggle raw display
$61$ \( T^{4} - 22 T^{3} + 136 T^{2} + \cdots - 832 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} - 154 T^{2} + \cdots + 4196 \) Copy content Toggle raw display
$71$ \( T^{4} + 14 T^{3} - 20 T^{2} + \cdots - 2304 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} - 40 T^{2} + 140 T + 536 \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} - 100 T^{2} + \cdots - 3392 \) Copy content Toggle raw display
$83$ \( T^{4} - 232 T^{2} + 944 T - 256 \) Copy content Toggle raw display
$89$ \( T^{4} + 44 T^{3} + 670 T^{2} + \cdots + 6360 \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} - 232 T^{2} + \cdots - 31072 \) Copy content Toggle raw display
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