Properties

Label 4056.2.a.v
Level $4056$
Weight $2$
Character orbit 4056.a
Self dual yes
Analytic conductor $32.387$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( - \beta + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + ( - \beta + 1) q^{7} + q^{9} + (\beta - 1) q^{11} - q^{15} + (\beta - 2) q^{17} + ( - \beta - 3) q^{19} + ( - \beta + 1) q^{21} + (\beta - 1) q^{23} - 4 q^{25} + q^{27} + 3 q^{29} + (2 \beta - 2) q^{31} + (\beta - 1) q^{33} + (\beta - 1) q^{35} + (\beta - 6) q^{37} + ( - \beta - 8) q^{41} + ( - \beta + 1) q^{43} - q^{45} + (\beta - 5) q^{47} + ( - 2 \beta + 7) q^{49} + (\beta - 2) q^{51} + 3 q^{53} + ( - \beta + 1) q^{55} + ( - \beta - 3) q^{57} + (2 \beta + 2) q^{59} + ( - \beta - 4) q^{61} + ( - \beta + 1) q^{63} + ( - \beta - 7) q^{67} + (\beta - 1) q^{69} + (\beta - 13) q^{71} + 7 q^{73} - 4 q^{75} + (2 \beta - 14) q^{77} + 12 q^{79} + q^{81} + ( - 3 \beta - 1) q^{83} + ( - \beta + 2) q^{85} + 3 q^{87} + ( - 4 \beta - 2) q^{89} + (2 \beta - 2) q^{93} + (\beta + 3) q^{95} + (2 \beta + 4) q^{97} + (\beta - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{15} - 4 q^{17} - 6 q^{19} + 2 q^{21} - 2 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 12 q^{37} - 16 q^{41} + 2 q^{43} - 2 q^{45} - 10 q^{47} + 14 q^{49} - 4 q^{51} + 6 q^{53} + 2 q^{55} - 6 q^{57} + 4 q^{59} - 8 q^{61} + 2 q^{63} - 14 q^{67} - 2 q^{69} - 26 q^{71} + 14 q^{73} - 8 q^{75} - 28 q^{77} + 24 q^{79} + 2 q^{81} - 2 q^{83} + 4 q^{85} + 6 q^{87} - 4 q^{89} - 4 q^{93} + 6 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 1.00000 0 −1.00000 0 −2.60555 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 4.60555 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.v 2
4.b odd 2 1 8112.2.a.bl 2
13.b even 2 1 4056.2.a.w 2
13.c even 3 2 312.2.q.d 4
13.d odd 4 2 4056.2.c.l 4
39.i odd 6 2 936.2.t.e 4
52.b odd 2 1 8112.2.a.bn 2
52.j odd 6 2 624.2.q.i 4
156.p even 6 2 1872.2.t.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.d 4 13.c even 3 2
624.2.q.i 4 52.j odd 6 2
936.2.t.e 4 39.i odd 6 2
1872.2.t.q 4 156.p even 6 2
4056.2.a.v 2 1.a even 1 1 trivial
4056.2.a.w 2 13.b even 2 1
4056.2.c.l 4 13.d odd 4 2
8112.2.a.bl 2 4.b odd 2 1
8112.2.a.bn 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 23 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 51 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$71$ \( T^{2} + 26T + 156 \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 116 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 204 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
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