Properties

Label 2200.4.a.b
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)
 
\(f(q)\) \(=\) \( q - 6 q^{3} + 32 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{3} + 32 q^{7} + 9 q^{9} + 11 q^{11} + 48 q^{13} + 36 q^{17} - 44 q^{19} - 192 q^{21} - 58 q^{23} + 108 q^{27} - 278 q^{29} - 112 q^{31} - 66 q^{33} - 194 q^{37} - 288 q^{39} - 314 q^{41} - 396 q^{43} + 410 q^{47} + 681 q^{49} - 216 q^{51} - 170 q^{53} + 264 q^{57} + 404 q^{59} + 250 q^{61} + 288 q^{63} + 26 q^{67} + 348 q^{69} - 468 q^{71} + 164 q^{73} + 352 q^{77} - 664 q^{79} - 891 q^{81} - 1348 q^{83} + 1668 q^{87} + 534 q^{89} + 1536 q^{91} + 672 q^{93} + 1498 q^{97} + 99 q^{99}+O(q^{100}) \) 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 −6.00000 0 0 0 32.0000 0 9.00000 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.b 1
5.b even 2 1 440.4.a.d 1
20.d odd 2 1 880.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.4.a.d 1 5.b even 2 1
880.4.a.d 1 20.d odd 2 1
2200.4.a.b 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2200))\):

\( T_{3} + 6 \) Copy content Toggle raw display
\( T_{7} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 6 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 32 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 48 \) Copy content Toggle raw display
$17$ \( T - 36 \) Copy content Toggle raw display
$19$ \( T + 44 \) Copy content Toggle raw display
$23$ \( T + 58 \) Copy content Toggle raw display
$29$ \( T + 278 \) Copy content Toggle raw display
$31$ \( T + 112 \) Copy content Toggle raw display
$37$ \( T + 194 \) Copy content Toggle raw display
$41$ \( T + 314 \) Copy content Toggle raw display
$43$ \( T + 396 \) Copy content Toggle raw display
$47$ \( T - 410 \) Copy content Toggle raw display
$53$ \( T + 170 \) Copy content Toggle raw display
$59$ \( T - 404 \) Copy content Toggle raw display
$61$ \( T - 250 \) Copy content Toggle raw display
$67$ \( T - 26 \) Copy content Toggle raw display
$71$ \( T + 468 \) Copy content Toggle raw display
$73$ \( T - 164 \) Copy content Toggle raw display
$79$ \( T + 664 \) Copy content Toggle raw display
$83$ \( T + 1348 \) Copy content Toggle raw display
$89$ \( T - 534 \) Copy content Toggle raw display
$97$ \( T - 1498 \) Copy content Toggle raw display
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