Properties

Label 4400.2.a.cb
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
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] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, 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("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} - q^{11} + 4 q^{13} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{17} + ( - \beta_{3} + 2 \beta_{2} + 2) q^{19} + ( - \beta_{3} + 2 \beta_1 - 2) q^{21} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{23} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{27} + (\beta_{3} + 2 \beta_1) q^{29} + (\beta_{3} - \beta_{2} - \beta_1 + 4) q^{31} + \beta_1 q^{33} + ( - 2 \beta_{2} - \beta_1 + 2) q^{37} - 4 \beta_1 q^{39} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{43} + (2 \beta_{2} + 2) q^{47} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 3) q^{49} + (\beta_{3} - 2 \beta_1 + 6) q^{51} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{53} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{57} + (2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{59} + ( - \beta_{3} - 2 \beta_1 + 8) q^{61} + ( - 2 \beta_{2} + 2 \beta_1 - 10) q^{63} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{67} + ( - 2 \beta_{3} - 3 \beta_{2} + \cdots - 6) q^{69}+ \cdots + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - q^{7} + 7 q^{9} - 4 q^{11} + 16 q^{13} + 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} + 15 q^{31} + q^{33} + 5 q^{37} - 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49} + 23 q^{51} + 5 q^{53} - 15 q^{57} - 6 q^{59} + 29 q^{61} - 40 q^{63} - 6 q^{67} - 30 q^{69} + q^{71} + 18 q^{73} + q^{77} + 20 q^{79} + 44 q^{81} - 26 q^{83} - 31 q^{87} + 21 q^{89} - 4 q^{91} + 25 q^{93} - 4 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 9\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 7\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 9\beta _1 + 2 \) 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
3.36007
0.655762
−0.339102
−2.67673
0 −3.36007 0 0 0 −1.08258 0 8.29009 0
1.2 0 −0.655762 0 0 0 0.415806 0 −2.56998 0
1.3 0 0.339102 0 0 0 4.05237 0 −2.88501 0
1.4 0 2.67673 0 0 0 −4.38559 0 4.16490 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)

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 4400.2.a.cb 4
4.b odd 2 1 2200.2.a.y 4
5.b even 2 1 4400.2.a.ce 4
5.c odd 4 2 880.2.b.j 8
20.d odd 2 1 2200.2.a.x 4
20.e even 4 2 440.2.b.d 8
60.l odd 4 2 3960.2.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.d 8 20.e even 4 2
880.2.b.j 8 5.c odd 4 2
2200.2.a.x 4 20.d odd 2 1
2200.2.a.y 4 4.b odd 2 1
3960.2.d.f 8 60.l odd 4 2
4400.2.a.cb 4 1.a even 1 1 trivial
4400.2.a.ce 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3}^{4} + T_{3}^{3} - 9T_{3}^{2} - 3T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} - 18T_{7}^{2} - 12T_{7} + 8 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 9 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 18 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T - 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$19$ \( T^{4} - 9 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots + 856 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$31$ \( T^{4} - 15 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 5 T^{3} + \cdots - 148 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + \cdots - 1024 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + \cdots + 1136 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + \cdots - 640 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} + \cdots + 1280 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$61$ \( T^{4} - 29 T^{3} + \cdots + 160 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} + \cdots + 568 \) Copy content Toggle raw display
$71$ \( T^{4} - T^{3} + \cdots - 1336 \) Copy content Toggle raw display
$73$ \( T^{4} - 18 T^{3} + \cdots - 1024 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} + \cdots - 19456 \) Copy content Toggle raw display
$83$ \( T^{4} + 26 T^{3} + \cdots - 6176 \) Copy content Toggle raw display
$89$ \( T^{4} - 21 T^{3} + \cdots - 346 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + \cdots + 512 \) Copy content Toggle raw display
show more
show less