Properties

Label 4992.2.a.s
Level $4992$
Weight $2$
Character orbit 4992.a
Self dual yes
Analytic conductor $39.861$
Analytic rank $0$
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] = [4992,2,Mod(1,4992)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4992, 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("4992.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4992 = 2^{7} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4992.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: \(39.8613206890\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_{2} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_{2} + 1) q^{7} + q^{9} + ( - \beta_{2} + \beta_1 + 2) q^{11} + q^{13} + (\beta_{2} - 1) q^{15} + (2 \beta_1 - 4) q^{17} + (\beta_{2} + 1) q^{19} + (\beta_{2} - 1) q^{21} + ( - \beta_{2} + 3 \beta_1 + 2) q^{23} + ( - 3 \beta_{2} - \beta_1 + 3) q^{25} - q^{27} + (\beta_{2} + 3 \beta_1 - 4) q^{29} + (\beta_{2} - 2 \beta_1 + 1) q^{31} + (\beta_{2} - \beta_1 - 2) q^{33} + ( - 3 \beta_{2} - \beta_1 + 8) q^{35} + (2 \beta_{2} + 2 \beta_1 + 2) q^{37} - q^{39} + ( - 2 \beta_{2} - \beta_1 - 3) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{43} + ( - \beta_{2} + 1) q^{45} - 2 q^{47} + ( - 3 \beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_1 + 4) q^{51} + ( - \beta_{2} - 3 \beta_1) q^{53} + ( - 4 \beta_{2} - 2 \beta_1 + 10) q^{55} + ( - \beta_{2} - 1) q^{57} + ( - \beta_{2} + \beta_1 + 2) q^{59} + ( - \beta_{2} - 5 \beta_1 + 4) q^{61} + ( - \beta_{2} + 1) q^{63} + ( - \beta_{2} + 1) q^{65} + (\beta_{2} + 6 \beta_1 - 5) q^{67} + (\beta_{2} - 3 \beta_1 - 2) q^{69} + (2 \beta_{2} - 2 \beta_1 + 2) q^{71} + (2 \beta_{2} + 2 \beta_1 - 10) q^{73} + (3 \beta_{2} + \beta_1 - 3) q^{75} + ( - 4 \beta_{2} - 2 \beta_1 + 10) q^{77} + (2 \beta_{2} + 4 \beta_1 - 4) q^{79} + q^{81} + (\beta_{2} + \beta_1 + 12) q^{83} + (4 \beta_{2} - 2 \beta_1 - 2) q^{85} + ( - \beta_{2} - 3 \beta_1 + 4) q^{87} + (3 \beta_1 - 9) q^{89} + ( - \beta_{2} + 1) q^{91} + ( - \beta_{2} + 2 \beta_1 - 1) q^{93} + (\beta_{2} + \beta_1 - 6) q^{95} + (4 \beta_{2} + 2 \beta_1 - 4) q^{97} + ( - \beta_{2} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + 6 q^{11} + 3 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} - 2 q^{21} + 8 q^{23} + 5 q^{25} - 3 q^{27} - 8 q^{29} + 2 q^{31} - 6 q^{33} + 20 q^{35} + 10 q^{37} - 3 q^{39} - 12 q^{41} + 12 q^{43} + 2 q^{45} - 6 q^{47} - q^{49} + 10 q^{51} - 4 q^{53} + 24 q^{55} - 4 q^{57} + 6 q^{59} + 6 q^{61} + 2 q^{63} + 2 q^{65} - 8 q^{67} - 8 q^{69} + 6 q^{71} - 26 q^{73} - 5 q^{75} + 24 q^{77} - 6 q^{79} + 3 q^{81} + 38 q^{83} - 4 q^{85} + 8 q^{87} - 24 q^{89} + 2 q^{91} - 2 q^{93} - 16 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
0.311108
2.17009
−1.48119
0 −1.00000 0 −1.52543 0 −1.52543 0 1.00000 0
1.2 0 −1.00000 0 −0.630898 0 −0.630898 0 1.00000 0
1.3 0 −1.00000 0 4.15633 0 4.15633 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4992.2.a.s yes 3
4.b odd 2 1 4992.2.a.z yes 3
8.b even 2 1 4992.2.a.w yes 3
8.d odd 2 1 4992.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4992.2.a.n 3 8.d odd 2 1
4992.2.a.s yes 3 1.a even 1 1 trivial
4992.2.a.w yes 3 8.b even 2 1
4992.2.a.z yes 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4992))\):

\( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 6T_{11}^{2} - 4T_{11} + 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 8 T - 4 \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 8 T - 4 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} - 4 T + 40 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 10 T^{2} + 12 T - 40 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 4 T + 20 \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} - 40 T + 304 \) Copy content Toggle raw display
$29$ \( T^{3} + 8 T^{2} - 32 T - 272 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 32 T - 52 \) Copy content Toggle raw display
$37$ \( T^{3} - 10 T^{2} - 20 T + 136 \) Copy content Toggle raw display
$41$ \( T^{3} + 12 T^{2} + 8 T - 172 \) Copy content Toggle raw display
$43$ \( T^{3} - 12 T^{2} - 64 T + 800 \) Copy content Toggle raw display
$47$ \( (T + 2)^{3} \) Copy content Toggle raw display
$53$ \( T^{3} + 4 T^{2} - 48 T + 80 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} - 4 T + 40 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 124 T + 760 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} - 172 T - 1252 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 52 T - 8 \) Copy content Toggle raw display
$73$ \( T^{3} + 26 T^{2} + 172 T + 184 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} - 100 T - 632 \) Copy content Toggle raw display
$83$ \( T^{3} - 38 T^{2} + 468 T - 1864 \) Copy content Toggle raw display
$89$ \( T^{3} + 24 T^{2} + 144 T + 108 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 148 T + 296 \) Copy content Toggle raw display
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