Properties

Label 2200.4.a.f
Level $2200$
Weight $4$
Character orbit 2200.a
Self dual yes
Analytic conductor $129.804$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 6 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 6 q^{7} - 26 q^{9} - 11 q^{11} + 40 q^{13} + 78 q^{17} + 36 q^{19} + 6 q^{21} - 7 q^{23} - 53 q^{27} + 8 q^{29} + 183 q^{31} - 11 q^{33} - 227 q^{37} + 40 q^{39} - 36 q^{41} - 322 q^{43} + 184 q^{47} - 307 q^{49} + 78 q^{51} + 6 q^{53} + 36 q^{57} - 99 q^{59} + 164 q^{61} - 156 q^{63} + 695 q^{67} - 7 q^{69} - 987 q^{71} + 248 q^{73} - 66 q^{77} - 242 q^{79} + 649 q^{81} + 1494 q^{83} + 8 q^{87} - 905 q^{89} + 240 q^{91} + 183 q^{93} + 1031 q^{97} + 286 q^{99}+O(q^{100}) \) 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 1.00000 0 0 0 6.00000 0 −26.0000 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.f 1
5.b even 2 1 88.4.a.a 1
15.d odd 2 1 792.4.a.e 1
20.d odd 2 1 176.4.a.d 1
40.e odd 2 1 704.4.a.f 1
40.f even 2 1 704.4.a.h 1
55.d odd 2 1 968.4.a.d 1
60.h even 2 1 1584.4.a.o 1
220.g even 2 1 1936.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.a.a 1 5.b even 2 1
176.4.a.d 1 20.d odd 2 1
704.4.a.f 1 40.e odd 2 1
704.4.a.h 1 40.f even 2 1
792.4.a.e 1 15.d odd 2 1
968.4.a.d 1 55.d odd 2 1
1584.4.a.o 1 60.h even 2 1
1936.4.a.i 1 220.g even 2 1
2200.4.a.f 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2200))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 6 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 40 \) Copy content Toggle raw display
$17$ \( T - 78 \) Copy content Toggle raw display
$19$ \( T - 36 \) Copy content Toggle raw display
$23$ \( T + 7 \) Copy content Toggle raw display
$29$ \( T - 8 \) Copy content Toggle raw display
$31$ \( T - 183 \) Copy content Toggle raw display
$37$ \( T + 227 \) Copy content Toggle raw display
$41$ \( T + 36 \) Copy content Toggle raw display
$43$ \( T + 322 \) Copy content Toggle raw display
$47$ \( T - 184 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 99 \) Copy content Toggle raw display
$61$ \( T - 164 \) Copy content Toggle raw display
$67$ \( T - 695 \) Copy content Toggle raw display
$71$ \( T + 987 \) Copy content Toggle raw display
$73$ \( T - 248 \) Copy content Toggle raw display
$79$ \( T + 242 \) Copy content Toggle raw display
$83$ \( T - 1494 \) Copy content Toggle raw display
$89$ \( T + 905 \) Copy content Toggle raw display
$97$ \( T - 1031 \) Copy content Toggle raw display
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