Properties

Label 400.4.a.j
Level $400$
Weight $4$
Character orbit 400.a
Self dual yes
Analytic conductor $23.601$
Analytic rank $1$
Dimension $1$
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] = [400,4,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, 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("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(23.6007640023\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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 \( q - q^{3} + 6 q^{7} - 26 q^{9} + 19 q^{11} + 12 q^{13} - 75 q^{17} + 91 q^{19} - 6 q^{21} - 174 q^{23} + 53 q^{27} - 272 q^{29} + 230 q^{31} - 19 q^{33} - 182 q^{37} - 12 q^{39} + 117 q^{41} - 372 q^{43} + 52 q^{47} - 307 q^{49} + 75 q^{51} - 402 q^{53} - 91 q^{57} - 312 q^{59} + 170 q^{61} - 156 q^{63} - 763 q^{67} + 174 q^{69} + 52 q^{71} - 981 q^{73} + 114 q^{77} - 1054 q^{79} + 649 q^{81} - 351 q^{83} + 272 q^{87} + 799 q^{89} + 72 q^{91} - 230 q^{93} + 962 q^{97} - 494 q^{99}+O(q^{100}) \) 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
\(2\) \( +1 \)
\(5\) \( -1 \)

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 400.4.a.j 1
4.b odd 2 1 200.4.a.f yes 1
5.b even 2 1 400.4.a.k 1
5.c odd 4 2 400.4.c.m 2
8.b even 2 1 1600.4.a.be 1
8.d odd 2 1 1600.4.a.w 1
12.b even 2 1 1800.4.a.l 1
20.d odd 2 1 200.4.a.e 1
20.e even 4 2 200.4.c.g 2
40.e odd 2 1 1600.4.a.bf 1
40.f even 2 1 1600.4.a.v 1
60.h even 2 1 1800.4.a.w 1
60.l odd 4 2 1800.4.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.e 1 20.d odd 2 1
200.4.a.f yes 1 4.b odd 2 1
200.4.c.g 2 20.e even 4 2
400.4.a.j 1 1.a even 1 1 trivial
400.4.a.k 1 5.b even 2 1
400.4.c.m 2 5.c odd 4 2
1600.4.a.v 1 40.f even 2 1
1600.4.a.w 1 8.d odd 2 1
1600.4.a.be 1 8.b even 2 1
1600.4.a.bf 1 40.e odd 2 1
1800.4.a.l 1 12.b even 2 1
1800.4.a.w 1 60.h even 2 1
1800.4.f.p 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(400))\):

\( T_{3} + 1 \) 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
$11$ \( T - 19 \) Copy content Toggle raw display
$13$ \( T - 12 \) Copy content Toggle raw display
$17$ \( T + 75 \) Copy content Toggle raw display
$19$ \( T - 91 \) Copy content Toggle raw display
$23$ \( T + 174 \) Copy content Toggle raw display
$29$ \( T + 272 \) Copy content Toggle raw display
$31$ \( T - 230 \) Copy content Toggle raw display
$37$ \( T + 182 \) Copy content Toggle raw display
$41$ \( T - 117 \) Copy content Toggle raw display
$43$ \( T + 372 \) Copy content Toggle raw display
$47$ \( T - 52 \) Copy content Toggle raw display
$53$ \( T + 402 \) Copy content Toggle raw display
$59$ \( T + 312 \) Copy content Toggle raw display
$61$ \( T - 170 \) Copy content Toggle raw display
$67$ \( T + 763 \) Copy content Toggle raw display
$71$ \( T - 52 \) Copy content Toggle raw display
$73$ \( T + 981 \) Copy content Toggle raw display
$79$ \( T + 1054 \) Copy content Toggle raw display
$83$ \( T + 351 \) Copy content Toggle raw display
$89$ \( T - 799 \) Copy content Toggle raw display
$97$ \( T - 962 \) Copy content Toggle raw display
show more
show less