Properties

Label 800.2.a.l
Level $800$
Weight $2$
Character orbit 800.a
Self dual yes
Analytic conductor $6.388$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 2 \beta q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 2 \beta q^{7} + 2 q^{9} - \beta q^{11} + 4 q^{13} - 7 q^{17} + 3 \beta q^{19} + 10 q^{21} + 2 \beta q^{23} + \beta q^{27} - 2 \beta q^{31} + 5 q^{33} + 2 q^{37} - 4 \beta q^{39} + 5 q^{41} + 4 \beta q^{47} + 13 q^{49} + 7 \beta q^{51} + 6 q^{53} - 15 q^{57} - 4 \beta q^{59} + 10 q^{61} - 4 \beta q^{63} + \beta q^{67} - 10 q^{69} + 4 \beta q^{71} - 9 q^{73} + 10 q^{77} + 2 \beta q^{79} - 11 q^{81} + 5 \beta q^{83} - 5 q^{89} - 8 \beta q^{91} + 10 q^{93} + 2 q^{97} - 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 8 q^{13} - 14 q^{17} + 20 q^{21} + 10 q^{33} + 4 q^{37} + 10 q^{41} + 26 q^{49} + 12 q^{53} - 30 q^{57} + 20 q^{61} - 20 q^{69} - 18 q^{73} + 20 q^{77} - 22 q^{81} - 10 q^{89} + 20 q^{93} + 4 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 −2.23607 0 0 0 −4.47214 0 2.00000 0
1.2 0 2.23607 0 0 0 4.47214 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.a.l yes 2
3.b odd 2 1 7200.2.a.cn 2
4.b odd 2 1 inner 800.2.a.l yes 2
5.b even 2 1 800.2.a.k 2
5.c odd 4 2 800.2.c.g 4
8.b even 2 1 1600.2.a.ba 2
8.d odd 2 1 1600.2.a.ba 2
12.b even 2 1 7200.2.a.cn 2
15.d odd 2 1 7200.2.a.cf 2
15.e even 4 2 7200.2.f.bg 4
20.d odd 2 1 800.2.a.k 2
20.e even 4 2 800.2.c.g 4
40.e odd 2 1 1600.2.a.bb 2
40.f even 2 1 1600.2.a.bb 2
40.i odd 4 2 1600.2.c.o 4
40.k even 4 2 1600.2.c.o 4
60.h even 2 1 7200.2.a.cf 2
60.l odd 4 2 7200.2.f.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 5.b even 2 1
800.2.a.k 2 20.d odd 2 1
800.2.a.l yes 2 1.a even 1 1 trivial
800.2.a.l yes 2 4.b odd 2 1 inner
800.2.c.g 4 5.c odd 4 2
800.2.c.g 4 20.e even 4 2
1600.2.a.ba 2 8.b even 2 1
1600.2.a.ba 2 8.d odd 2 1
1600.2.a.bb 2 40.e odd 2 1
1600.2.a.bb 2 40.f even 2 1
1600.2.c.o 4 40.i odd 4 2
1600.2.c.o 4 40.k even 4 2
7200.2.a.cf 2 15.d odd 2 1
7200.2.a.cf 2 60.h even 2 1
7200.2.a.cn 2 3.b odd 2 1
7200.2.a.cn 2 12.b even 2 1
7200.2.f.bg 4 15.e even 4 2
7200.2.f.bg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} - 5 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20 \) Copy content Toggle raw display
$11$ \( T^{2} - 5 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 45 \) Copy content Toggle raw display
$23$ \( T^{2} - 20 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 20 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 80 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 80 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 5 \) Copy content Toggle raw display
$71$ \( T^{2} - 80 \) Copy content Toggle raw display
$73$ \( (T + 9)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 20 \) Copy content Toggle raw display
$83$ \( T^{2} - 125 \) Copy content Toggle raw display
$89$ \( (T + 5)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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