Properties

Label 800.4.a.i
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 5 q^{3} - 10 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{3} - 10 q^{7} - 2 q^{9} - 15 q^{11} - 8 q^{13} + 21 q^{17} + 105 q^{19} - 50 q^{21} - 10 q^{23} - 145 q^{27} - 20 q^{29} - 230 q^{31} - 75 q^{33} + 54 q^{37} - 40 q^{39} - 195 q^{41} - 300 q^{43} - 480 q^{47} - 243 q^{49} + 105 q^{51} - 322 q^{53} + 525 q^{57} + 560 q^{59} - 730 q^{61} + 20 q^{63} + 255 q^{67} - 50 q^{69} - 40 q^{71} - 317 q^{73} + 150 q^{77} - 830 q^{79} - 671 q^{81} + 75 q^{83} - 100 q^{87} - 705 q^{89} + 80 q^{91} - 1150 q^{93} + 1434 q^{97} + 30 q^{99}+O(q^{100}) \) 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 −10.0000 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 800.4.a.i yes 1
4.b odd 2 1 800.4.a.c yes 1
5.b even 2 1 800.4.a.b 1
5.c odd 4 2 800.4.c.c 2
8.b even 2 1 1600.4.a.l 1
8.d odd 2 1 1600.4.a.bp 1
20.d odd 2 1 800.4.a.j yes 1
20.e even 4 2 800.4.c.d 2
40.e odd 2 1 1600.4.a.k 1
40.f even 2 1 1600.4.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.b 1 5.b even 2 1
800.4.a.c yes 1 4.b odd 2 1
800.4.a.i yes 1 1.a even 1 1 trivial
800.4.a.j yes 1 20.d odd 2 1
800.4.c.c 2 5.c odd 4 2
800.4.c.d 2 20.e even 4 2
1600.4.a.k 1 40.e odd 2 1
1600.4.a.l 1 8.b even 2 1
1600.4.a.bp 1 8.d odd 2 1
1600.4.a.bq 1 40.f even 2 1

Hecke kernels

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

\( T_{3} - 5 \) Copy content Toggle raw display
\( T_{11} + 15 \) Copy content Toggle raw display
\( T_{13} + 8 \) 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
$7$ \( T + 10 \) Copy content Toggle raw display
$11$ \( T + 15 \) Copy content Toggle raw display
$13$ \( T + 8 \) Copy content Toggle raw display
$17$ \( T - 21 \) Copy content Toggle raw display
$19$ \( T - 105 \) Copy content Toggle raw display
$23$ \( T + 10 \) Copy content Toggle raw display
$29$ \( T + 20 \) Copy content Toggle raw display
$31$ \( T + 230 \) Copy content Toggle raw display
$37$ \( T - 54 \) Copy content Toggle raw display
$41$ \( T + 195 \) Copy content Toggle raw display
$43$ \( T + 300 \) Copy content Toggle raw display
$47$ \( T + 480 \) Copy content Toggle raw display
$53$ \( T + 322 \) Copy content Toggle raw display
$59$ \( T - 560 \) Copy content Toggle raw display
$61$ \( T + 730 \) Copy content Toggle raw display
$67$ \( T - 255 \) Copy content Toggle raw display
$71$ \( T + 40 \) Copy content Toggle raw display
$73$ \( T + 317 \) Copy content Toggle raw display
$79$ \( T + 830 \) Copy content Toggle raw display
$83$ \( T - 75 \) Copy content Toggle raw display
$89$ \( T + 705 \) Copy content Toggle raw display
$97$ \( T - 1434 \) Copy content Toggle raw display
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