↑

Properties

Label 4056.2.a.bi
Level $4056$
Weight $2$
Character orbit 4056.a
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.27700337.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{3} - \beta_1) q^{5} + (\beta_{4} + \beta_{2} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_{3} - \beta_1) q^{5} + (\beta_{4} + \beta_{2} - 1) q^{7} + q^{9} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots - 2) q^{11}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9} - 6 q^{11} + q^{15} + 9 q^{17} + 7 q^{19} - 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} - 11 q^{31} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 13 q^{41} + 15 q^{43} + q^{45} - 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} + 7 q^{57} - 7 q^{59} + 25 q^{61} - 5 q^{63} + 5 q^{67} + 12 q^{69} + 8 q^{71} + 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} - 13 q^{83} - 35 q^{85} + 7 q^{87} + 33 q^{89} - 11 q^{93} + 47 q^{95} + 50 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 11\nu^{4} + 15\nu^{3} - 136\nu^{2} + 160\nu + 338 ) / 167 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{5} - 17\nu^{4} + 159\nu^{3} + 195\nu^{2} - 642\nu - 325 ) / 167 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\nu^{5} + 56\nu^{4} - 121\nu^{3} - 662\nu^{2} + 268\nu + 1326 ) / 167 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{5} - 6\nu^{4} + 174\nu^{3} + 226\nu^{2} - 482\nu - 1156 ) / 167 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} - \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} - 2\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{5} + 3\beta_{4} - 10\beta_{3} - 9\beta_{2} + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{5} + 12\beta_{4} + 14\beta_{3} - 40\beta_{2} + 70\beta _1 + 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.88227
3.16419
2.72245
−3.12925
−0.920510
−2.71914
0 1.00000 0 −3.12925 0 −1.48962 0 1.00000 0
1.2 0 1.00000 0 −2.71914 0 0.735542 0 1.00000 0
1.3 0 1.00000 0 −0.920510 0 −4.87031 0 1.00000 0
1.4 0 1.00000 0 1.88227 0 1.29156 0 1.00000 0
1.5 0 1.00000 0 2.72245 0 3.31535 0 1.00000 0
1.6 0 1.00000 0 3.16419 0 −3.98252 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bi yes 6
4.b odd 2 1 8112.2.a.cu 6
13.b even 2 1 4056.2.a.bh ✓ 6
13.d odd 4 2 4056.2.c.r 12
52.b odd 2 1 8112.2.a.ct 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4056.2.a.bh ✓ 6 13.b even 2 1
4056.2.a.bi yes 6 1.a even 1 1 trivial
4056.2.c.r 12 13.d odd 4 2
8112.2.a.ct 6 52.b odd 2 1
8112.2.a.cu 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\):

\( T_{5}^{6} - T_{5}^{5} - 19T_{5}^{4} + 17T_{5}^{3} + 103T_{5}^{2} - 71T_{5} - 127 \) Copy content Toggle raw display
\( T_{7}^{6} + 5T_{7}^{5} - 15T_{7}^{4} - 69T_{7}^{3} + 63T_{7}^{2} + 119T_{7} - 91 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + \cdots - 127 \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + \cdots - 91 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots - 91 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 9 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$19$ \( T^{6} - 7 T^{5} + \cdots + 832 \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} + \cdots + 16856 \) Copy content Toggle raw display
$29$ \( T^{6} - 7 T^{5} + \cdots + 27397 \) Copy content Toggle raw display
$31$ \( T^{6} + 11 T^{5} + \cdots - 301 \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + \cdots + 664 \) Copy content Toggle raw display
$41$ \( T^{6} - 13 T^{5} + \cdots - 78728 \) Copy content Toggle raw display
$43$ \( T^{6} - 15 T^{5} + \cdots - 249992 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + \cdots + 287552 \) Copy content Toggle raw display
$53$ \( (T^{3} - 11 T^{2} + \cdots - 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 7 T^{5} + \cdots + 1112 \) Copy content Toggle raw display
$61$ \( T^{6} - 25 T^{5} + \cdots + 22504 \) Copy content Toggle raw display
$67$ \( T^{6} - 5 T^{5} + \cdots + 7112 \) Copy content Toggle raw display
$71$ \( T^{6} - 8 T^{5} + \cdots - 23192 \) Copy content Toggle raw display
$73$ \( T^{6} - 15 T^{5} + \cdots + 109067 \) Copy content Toggle raw display
$79$ \( T^{6} - 14 T^{5} + \cdots + 784147 \) Copy content Toggle raw display
$83$ \( T^{6} + 13 T^{5} + \cdots + 598949 \) Copy content Toggle raw display
$89$ \( T^{6} - 33 T^{5} + \cdots + 786344 \) Copy content Toggle raw display
$97$ \( T^{6} - 50 T^{5} + \cdots + 711061 \) Copy content Toggle raw display
show more
show less