Properties

Label 2200.4.a.i
Level $2200$
Weight $4$
Character orbit 2200.a
Self dual yes
Analytic conductor $129.804$
Analytic rank $1$
Dimension $1$
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] = [2200,4,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(129.804202013\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 5 q^{3} - q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{3} - q^{7} - 2 q^{9} + 11 q^{11} - 18 q^{13} + 113 q^{17} + 55 q^{19} - 5 q^{21} - 190 q^{23} - 145 q^{27} - 69 q^{29} - 255 q^{31} + 55 q^{33} - 51 q^{37} - 90 q^{39} - 314 q^{41} + 484 q^{43} - 470 q^{47} - 342 q^{49} + 565 q^{51} + 545 q^{53} + 275 q^{57} - 102 q^{59} + 129 q^{61} + 2 q^{63} + 664 q^{67} - 950 q^{69} - 1029 q^{71} + 758 q^{73} - 11 q^{77} + 634 q^{79} - 671 q^{81} + 654 q^{83} - 345 q^{87} - 511 q^{89} + 18 q^{91} - 1275 q^{93} - 1736 q^{97} - 22 q^{99}+O(q^{100}) \) 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
0
0 5.00000 0 0 0 −1.00000 0 −2.00000 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 2200.4.a.i 1
5.b even 2 1 440.4.a.a 1
20.d odd 2 1 880.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.4.a.a 1 5.b even 2 1
880.4.a.n 1 20.d odd 2 1
2200.4.a.i 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2200))\):

\( T_{3} - 5 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 5 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T + 18 \) Copy content Toggle raw display
$17$ \( T - 113 \) Copy content Toggle raw display
$19$ \( T - 55 \) Copy content Toggle raw display
$23$ \( T + 190 \) Copy content Toggle raw display
$29$ \( T + 69 \) Copy content Toggle raw display
$31$ \( T + 255 \) Copy content Toggle raw display
$37$ \( T + 51 \) Copy content Toggle raw display
$41$ \( T + 314 \) Copy content Toggle raw display
$43$ \( T - 484 \) Copy content Toggle raw display
$47$ \( T + 470 \) Copy content Toggle raw display
$53$ \( T - 545 \) Copy content Toggle raw display
$59$ \( T + 102 \) Copy content Toggle raw display
$61$ \( T - 129 \) Copy content Toggle raw display
$67$ \( T - 664 \) Copy content Toggle raw display
$71$ \( T + 1029 \) Copy content Toggle raw display
$73$ \( T - 758 \) Copy content Toggle raw display
$79$ \( T - 634 \) Copy content Toggle raw display
$83$ \( T - 654 \) Copy content Toggle raw display
$89$ \( T + 511 \) Copy content Toggle raw display
$97$ \( T + 1736 \) Copy content Toggle raw display
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