Properties

Label 4056.2.a.x
Level $4056$
Weight $2$
Character orbit 4056.a
Self dual yes
Analytic conductor $32.387$
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] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + q^{9} + (\beta_{2} - \beta_1 + 1) q^{11} + (\beta_{2} + 1) q^{15} - \beta_1 q^{17} + (2 \beta_{2} - \beta_1 + 2) q^{19} + (\beta_{2} - 2 \beta_1 + 1) q^{21} + ( - \beta_{2} - 1) q^{23} + (\beta_{2} + \beta_1 - 3) q^{25} - q^{27} + (\beta_{2} + 3 \beta_1 - 2) q^{29} + ( - \beta_{2} - 5 \beta_1 + 2) q^{31} + ( - \beta_{2} + \beta_1 - 1) q^{33} + ( - \beta_{2} - \beta_1) q^{35} + (4 \beta_{2} - 3 \beta_1 - 1) q^{37} + (5 \beta_{2} + \beta_1 + 1) q^{41} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{43} + ( - \beta_{2} - 1) q^{45} + (4 \beta_{2} - 3 \beta_1 + 8) q^{47} + (\beta_{2} - 3 \beta_1 - 1) q^{49} + \beta_1 q^{51} + (5 \beta_{2} - \beta_1 + 7) q^{53} - q^{55} + ( - 2 \beta_{2} + \beta_1 - 2) q^{57} + ( - 6 \beta_{2} + 8 \beta_1) q^{59} + ( - 6 \beta_{2} + 7 \beta_1 - 6) q^{61} + ( - \beta_{2} + 2 \beta_1 - 1) q^{63} + (3 \beta_{2} - 4 \beta_1 + 10) q^{67} + (\beta_{2} + 1) q^{69} + (3 \beta_{2} + \beta_1 - 2) q^{71} + (\beta_{2} - 2 \beta_1 - 1) q^{73} + ( - \beta_{2} - \beta_1 + 3) q^{75} + (2 \beta_1 - 3) q^{77} + (3 \beta_{2} - 8 \beta_1 - 2) q^{79} + q^{81} + ( - \beta_1 + 5) q^{83} + (\beta_{2} + \beta_1 + 1) q^{85} + ( - \beta_{2} - 3 \beta_1 + 2) q^{87} + (\beta_{2} + 4 \beta_1 - 8) q^{89} + (\beta_{2} + 5 \beta_1 - 2) q^{93} + ( - \beta_{2} - \beta_1 - 3) q^{95} + ( - 2 \beta_{2} - 13) q^{97} + (\beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 2 q^{5} + 3 q^{9} + q^{11} + 2 q^{15} - q^{17} + 3 q^{19} - 2 q^{23} - 9 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 10 q^{37} - q^{41} - 9 q^{43} - 2 q^{45} + 17 q^{47} - 7 q^{49} + q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 14 q^{59} - 5 q^{61} + 23 q^{67} + 2 q^{69} - 8 q^{71} - 6 q^{73} + 9 q^{75} - 7 q^{77} - 17 q^{79} + 3 q^{81} + 14 q^{83} + 3 q^{85} + 4 q^{87} - 21 q^{89} - 2 q^{93} - 9 q^{95} - 37 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
−1.24698
0.445042
0 −1.00000 0 −2.24698 0 1.35690 0 1.00000 0
1.2 0 −1.00000 0 −0.554958 0 −3.04892 0 1.00000 0
1.3 0 −1.00000 0 0.801938 0 1.69202 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 4056.2.a.x 3
4.b odd 2 1 8112.2.a.ci 3
13.b even 2 1 4056.2.a.ba yes 3
13.d odd 4 2 4056.2.c.n 6
52.b odd 2 1 8112.2.a.cn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4056.2.a.x 3 1.a even 1 1 trivial
4056.2.a.ba yes 3 13.b even 2 1
4056.2.c.n 6 13.d odd 4 2
8112.2.a.ci 3 4.b odd 2 1
8112.2.a.cn 3 52.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\):

\( T_{5}^{3} + 2T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 7T_{7} + 7 \) 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
$5$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} - 7T + 7 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$23$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 25 T - 71 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 71 T + 113 \) Copy content Toggle raw display
$37$ \( T^{3} + 10 T^{2} + 3 T - 13 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} - 72 T - 169 \) Copy content Toggle raw display
$43$ \( T^{3} + 9T^{2} - T - 1 \) Copy content Toggle raw display
$47$ \( T^{3} - 17 T^{2} + 66 T + 41 \) Copy content Toggle raw display
$53$ \( T^{3} - 15 T^{2} + 26 T + 169 \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} - 56 T + 728 \) Copy content Toggle raw display
$61$ \( T^{3} + 5 T^{2} - 92 T - 83 \) Copy content Toggle raw display
$67$ \( T^{3} - 23 T^{2} + 146 T - 251 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} - 9 T - 113 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} + 5 T - 13 \) Copy content Toggle raw display
$79$ \( T^{3} + 17 T^{2} - 18 T - 923 \) Copy content Toggle raw display
$83$ \( T^{3} - 14 T^{2} + 63 T - 91 \) Copy content Toggle raw display
$89$ \( T^{3} + 21 T^{2} + 98 T - 49 \) Copy content Toggle raw display
$97$ \( T^{3} + 37 T^{2} + 447 T + 1763 \) Copy content Toggle raw display
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