Properties

Label 200.4.a.c
Level $200$
Weight $4$
Character orbit 200.a
Self dual yes
Analytic conductor $11.800$
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] = [200,4,Mod(1,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.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: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{3} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{3} - 2 q^{7} - 2 q^{9} + 39 q^{11} - 84 q^{13} + 61 q^{17} + 151 q^{19} + 10 q^{21} + 58 q^{23} + 145 q^{27} + 192 q^{29} - 18 q^{31} - 195 q^{33} + 138 q^{37} + 420 q^{39} + 229 q^{41} + 164 q^{43} + 212 q^{47} - 339 q^{49} - 305 q^{51} - 578 q^{53} - 755 q^{57} - 336 q^{59} + 858 q^{61} + 4 q^{63} + 209 q^{67} - 290 q^{69} - 780 q^{71} + 403 q^{73} - 78 q^{77} - 230 q^{79} - 671 q^{81} + 1293 q^{83} - 960 q^{87} - 1369 q^{89} + 168 q^{91} + 90 q^{93} - 382 q^{97} - 78 q^{99}+O(q^{100}) \) 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 −2.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 200.4.a.c 1
3.b odd 2 1 1800.4.a.p 1
4.b odd 2 1 400.4.a.q 1
5.b even 2 1 200.4.a.h yes 1
5.c odd 4 2 200.4.c.d 2
8.b even 2 1 1600.4.a.bn 1
8.d odd 2 1 1600.4.a.n 1
15.d odd 2 1 1800.4.a.t 1
15.e even 4 2 1800.4.f.c 2
20.d odd 2 1 400.4.a.f 1
20.e even 4 2 400.4.c.g 2
40.e odd 2 1 1600.4.a.bo 1
40.f even 2 1 1600.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.c 1 1.a even 1 1 trivial
200.4.a.h yes 1 5.b even 2 1
200.4.c.d 2 5.c odd 4 2
400.4.a.f 1 20.d odd 2 1
400.4.a.q 1 4.b odd 2 1
400.4.c.g 2 20.e even 4 2
1600.4.a.m 1 40.f even 2 1
1600.4.a.n 1 8.d odd 2 1
1600.4.a.bn 1 8.b even 2 1
1600.4.a.bo 1 40.e odd 2 1
1800.4.a.p 1 3.b odd 2 1
1800.4.a.t 1 15.d odd 2 1
1800.4.f.c 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(200))\). 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
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 39 \) Copy content Toggle raw display
$13$ \( T + 84 \) Copy content Toggle raw display
$17$ \( T - 61 \) Copy content Toggle raw display
$19$ \( T - 151 \) Copy content Toggle raw display
$23$ \( T - 58 \) Copy content Toggle raw display
$29$ \( T - 192 \) Copy content Toggle raw display
$31$ \( T + 18 \) Copy content Toggle raw display
$37$ \( T - 138 \) Copy content Toggle raw display
$41$ \( T - 229 \) Copy content Toggle raw display
$43$ \( T - 164 \) Copy content Toggle raw display
$47$ \( T - 212 \) Copy content Toggle raw display
$53$ \( T + 578 \) Copy content Toggle raw display
$59$ \( T + 336 \) Copy content Toggle raw display
$61$ \( T - 858 \) Copy content Toggle raw display
$67$ \( T - 209 \) Copy content Toggle raw display
$71$ \( T + 780 \) Copy content Toggle raw display
$73$ \( T - 403 \) Copy content Toggle raw display
$79$ \( T + 230 \) Copy content Toggle raw display
$83$ \( T - 1293 \) Copy content Toggle raw display
$89$ \( T + 1369 \) Copy content Toggle raw display
$97$ \( T + 382 \) Copy content Toggle raw display
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