Properties

Label 800.4.a.d
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
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] = [800,4,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, 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("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} + 6 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + 6 q^{7} - 23 q^{9} - 60 q^{11} - 50 q^{13} + 30 q^{17} - 40 q^{19} - 12 q^{21} + 178 q^{23} + 100 q^{27} + 166 q^{29} - 20 q^{31} + 120 q^{33} - 10 q^{37} + 100 q^{39} - 250 q^{41} + 142 q^{43} + 214 q^{47} - 307 q^{49} - 60 q^{51} - 490 q^{53} + 80 q^{57} + 800 q^{59} + 250 q^{61} - 138 q^{63} - 774 q^{67} - 356 q^{69} - 100 q^{71} + 230 q^{73} - 360 q^{77} + 1320 q^{79} + 421 q^{81} + 982 q^{83} - 332 q^{87} + 874 q^{89} - 300 q^{91} + 40 q^{93} + 310 q^{97} + 1380 q^{99}+O(q^{100}) \) 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 −2.00000 0 0 0 6.00000 0 −23.0000 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 800.4.a.d 1
4.b odd 2 1 800.4.a.h 1
5.b even 2 1 160.4.a.b yes 1
5.c odd 4 2 800.4.c.e 2
8.b even 2 1 1600.4.a.bj 1
8.d odd 2 1 1600.4.a.r 1
15.d odd 2 1 1440.4.a.n 1
20.d odd 2 1 160.4.a.a 1
20.e even 4 2 800.4.c.f 2
40.e odd 2 1 320.4.a.i 1
40.f even 2 1 320.4.a.f 1
60.h even 2 1 1440.4.a.o 1
80.k odd 4 2 1280.4.d.f 2
80.q even 4 2 1280.4.d.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 20.d odd 2 1
160.4.a.b yes 1 5.b even 2 1
320.4.a.f 1 40.f even 2 1
320.4.a.i 1 40.e odd 2 1
800.4.a.d 1 1.a even 1 1 trivial
800.4.a.h 1 4.b odd 2 1
800.4.c.e 2 5.c odd 4 2
800.4.c.f 2 20.e even 4 2
1280.4.d.f 2 80.k odd 4 2
1280.4.d.k 2 80.q even 4 2
1440.4.a.n 1 15.d odd 2 1
1440.4.a.o 1 60.h even 2 1
1600.4.a.r 1 8.d odd 2 1
1600.4.a.bj 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{11} + 60 \) Copy content Toggle raw display
\( T_{13} + 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 6 \) Copy content Toggle raw display
$11$ \( T + 60 \) Copy content Toggle raw display
$13$ \( T + 50 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T + 40 \) Copy content Toggle raw display
$23$ \( T - 178 \) Copy content Toggle raw display
$29$ \( T - 166 \) Copy content Toggle raw display
$31$ \( T + 20 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 250 \) Copy content Toggle raw display
$43$ \( T - 142 \) Copy content Toggle raw display
$47$ \( T - 214 \) Copy content Toggle raw display
$53$ \( T + 490 \) Copy content Toggle raw display
$59$ \( T - 800 \) Copy content Toggle raw display
$61$ \( T - 250 \) Copy content Toggle raw display
$67$ \( T + 774 \) Copy content Toggle raw display
$71$ \( T + 100 \) Copy content Toggle raw display
$73$ \( T - 230 \) Copy content Toggle raw display
$79$ \( T - 1320 \) Copy content Toggle raw display
$83$ \( T - 982 \) Copy content Toggle raw display
$89$ \( T - 874 \) Copy content Toggle raw display
$97$ \( T - 310 \) Copy content Toggle raw display
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