Properties

Label 400.2.a.e
Level $400$
Weight $2$
Character orbit 400.a
Self dual yes
Analytic conductor $3.194$
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] = [400,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.19401608085\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} + 4 q^{23} - 2 q^{29} + 8 q^{31} - 6 q^{37} - 6 q^{41} - 8 q^{43} + 4 q^{47} + 9 q^{49} - 6 q^{53} + 4 q^{59} - 2 q^{61} + 12 q^{63} + 8 q^{67} + 6 q^{73} + 16 q^{77} + 9 q^{81} - 16 q^{83} - 6 q^{89} - 8 q^{91} + 14 q^{97} + 12 q^{99}+O(q^{100}) \) 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 0 0 0 0 −4.00000 0 −3.00000 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 400.2.a.e 1
3.b odd 2 1 3600.2.a.h 1
4.b odd 2 1 200.2.a.c 1
5.b even 2 1 80.2.a.a 1
5.c odd 4 2 400.2.c.d 2
8.b even 2 1 1600.2.a.k 1
8.d odd 2 1 1600.2.a.o 1
12.b even 2 1 1800.2.a.v 1
15.d odd 2 1 720.2.a.e 1
15.e even 4 2 3600.2.f.t 2
20.d odd 2 1 40.2.a.a 1
20.e even 4 2 200.2.c.b 2
28.d even 2 1 9800.2.a.x 1
35.c odd 2 1 3920.2.a.s 1
40.e odd 2 1 320.2.a.c 1
40.f even 2 1 320.2.a.d 1
40.i odd 4 2 1600.2.c.m 2
40.k even 4 2 1600.2.c.k 2
55.d odd 2 1 9680.2.a.q 1
60.h even 2 1 360.2.a.a 1
60.l odd 4 2 1800.2.f.a 2
80.k odd 4 2 1280.2.d.j 2
80.q even 4 2 1280.2.d.a 2
120.i odd 2 1 2880.2.a.bg 1
120.m even 2 1 2880.2.a.t 1
140.c even 2 1 1960.2.a.g 1
140.p odd 6 2 1960.2.q.h 2
140.s even 6 2 1960.2.q.i 2
180.n even 6 2 3240.2.q.x 2
180.p odd 6 2 3240.2.q.k 2
220.g even 2 1 4840.2.a.f 1
260.g odd 2 1 6760.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 20.d odd 2 1
80.2.a.a 1 5.b even 2 1
200.2.a.c 1 4.b odd 2 1
200.2.c.b 2 20.e even 4 2
320.2.a.c 1 40.e odd 2 1
320.2.a.d 1 40.f even 2 1
360.2.a.a 1 60.h even 2 1
400.2.a.e 1 1.a even 1 1 trivial
400.2.c.d 2 5.c odd 4 2
720.2.a.e 1 15.d odd 2 1
1280.2.d.a 2 80.q even 4 2
1280.2.d.j 2 80.k odd 4 2
1600.2.a.k 1 8.b even 2 1
1600.2.a.o 1 8.d odd 2 1
1600.2.c.k 2 40.k even 4 2
1600.2.c.m 2 40.i odd 4 2
1800.2.a.v 1 12.b even 2 1
1800.2.f.a 2 60.l odd 4 2
1960.2.a.g 1 140.c even 2 1
1960.2.q.h 2 140.p odd 6 2
1960.2.q.i 2 140.s even 6 2
2880.2.a.t 1 120.m even 2 1
2880.2.a.bg 1 120.i odd 2 1
3240.2.q.k 2 180.p odd 6 2
3240.2.q.x 2 180.n even 6 2
3600.2.a.h 1 3.b odd 2 1
3600.2.f.t 2 15.e even 4 2
3920.2.a.s 1 35.c odd 2 1
4840.2.a.f 1 220.g even 2 1
6760.2.a.i 1 260.g odd 2 1
9680.2.a.q 1 55.d odd 2 1
9800.2.a.x 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 16 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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