Properties

Label 4056.2.c.k
Level $4056$
Weight $2$
Character orbit 4056.c
Analytic conductor $32.387$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(337,4056)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4056.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + q^{9} + (\beta_{2} - \beta_1) q^{11} + \beta_{2} q^{15} + (\beta_{3} + 2) q^{17} + (\beta_{2} - 3 \beta_1) q^{19} + ( - \beta_{2} + \beta_1) q^{21} + (3 \beta_{3} - 3) q^{23} + 2 q^{25} - q^{27} + ( - 4 \beta_{3} + 1) q^{29} + (2 \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} + \beta_1) q^{33} + ( - \beta_{3} + 3) q^{35} + ( - 2 \beta_{2} - 5 \beta_1) q^{37} + 5 \beta_1 q^{41} + (3 \beta_{3} - 7) q^{43} - \beta_{2} q^{45} + (\beta_{2} + 5 \beta_1) q^{47} + (2 \beta_{3} + 3) q^{49} + ( - \beta_{3} - 2) q^{51} + (4 \beta_{3} - 1) q^{53} + ( - \beta_{3} + 3) q^{55} + ( - \beta_{2} + 3 \beta_1) q^{57} + ( - \beta_{3} - 8) q^{61} + (\beta_{2} - \beta_1) q^{63} + ( - \beta_{2} + 7 \beta_1) q^{67} + ( - 3 \beta_{3} + 3) q^{69} + ( - 3 \beta_{2} - 3 \beta_1) q^{71} + (3 \beta_{2} - 4 \beta_1) q^{73} - 2 q^{75} + (2 \beta_{3} - 4) q^{77} + (6 \beta_{3} + 2) q^{79} + q^{81} + (5 \beta_{2} + 7 \beta_1) q^{83} + ( - 2 \beta_{2} - 3 \beta_1) q^{85} + (4 \beta_{3} - 1) q^{87} + ( - 2 \beta_{2} + 6 \beta_1) q^{89} + ( - 2 \beta_{2} - 2 \beta_1) q^{93} + ( - 3 \beta_{3} + 3) q^{95} - 10 \beta_1 q^{97} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{9} + 8 q^{17} - 12 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} + 12 q^{35} - 28 q^{43} + 12 q^{49} - 8 q^{51} - 4 q^{53} + 12 q^{55} - 32 q^{61} + 12 q^{69} - 8 q^{75} - 16 q^{77} + 8 q^{79} + 4 q^{81} - 4 q^{87} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −1.00000 0 1.73205i 0 0.732051i 0 1.00000 0
337.2 0 −1.00000 0 1.73205i 0 2.73205i 0 1.00000 0
337.3 0 −1.00000 0 1.73205i 0 2.73205i 0 1.00000 0
337.4 0 −1.00000 0 1.73205i 0 0.732051i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.k 4
13.b even 2 1 inner 4056.2.c.k 4
13.c even 3 1 312.2.bf.a 4
13.d odd 4 1 4056.2.a.t 2
13.d odd 4 1 4056.2.a.u 2
13.e even 6 1 312.2.bf.a 4
39.h odd 6 1 936.2.bi.a 4
39.i odd 6 1 936.2.bi.a 4
52.f even 4 1 8112.2.a.br 2
52.f even 4 1 8112.2.a.bw 2
52.i odd 6 1 624.2.bv.c 4
52.j odd 6 1 624.2.bv.c 4
156.p even 6 1 1872.2.by.g 4
156.r even 6 1 1872.2.by.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.a 4 13.c even 3 1
312.2.bf.a 4 13.e even 6 1
624.2.bv.c 4 52.i odd 6 1
624.2.bv.c 4 52.j odd 6 1
936.2.bi.a 4 39.h odd 6 1
936.2.bi.a 4 39.i odd 6 1
1872.2.by.g 4 156.p even 6 1
1872.2.by.g 4 156.r even 6 1
4056.2.a.t 2 13.d odd 4 1
4056.2.a.u 2 13.d odd 4 1
4056.2.c.k 4 1.a even 1 1 trivial
4056.2.c.k 4 13.b even 2 1 inner
8112.2.a.br 2 52.f even 4 1
8112.2.a.bw 2 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 47)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} + 74T^{2} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T + 22)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T - 47)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 16 T + 61)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$71$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$73$ \( T^{4} + 86T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 248T^{2} + 676 \) Copy content Toggle raw display
$89$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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