Properties

Label 800.2.a.n
Level $800$
Weight $2$
Character orbit 800.a
Self dual yes
Analytic conductor $6.388$
Analytic rank $0$
Dimension $2$
CM discriminant -20
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: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + ( - \beta + 3) q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + ( - \beta + 3) q^{7} + (2 \beta + 3) q^{9} + (2 \beta - 2) q^{21} + ( - 3 \beta + 1) q^{23} + (2 \beta + 10) q^{27} - 6 q^{29} - 2 \beta q^{41} + (\beta + 9) q^{43} + (3 \beta + 7) q^{47} + ( - 6 \beta + 7) q^{49} - 6 \beta q^{61} + (3 \beta - 1) q^{63} + ( - 5 \beta + 3) q^{67} + ( - 2 \beta - 14) q^{69} + (6 \beta + 11) q^{81} + ( - 3 \beta - 11) q^{83} + ( - 6 \beta - 6) q^{87} + 6 q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{7} + 6 q^{9} - 4 q^{21} + 2 q^{23} + 20 q^{27} - 12 q^{29} + 18 q^{43} + 14 q^{47} + 14 q^{49} - 2 q^{63} + 6 q^{67} - 28 q^{69} + 22 q^{81} - 22 q^{83} - 12 q^{87} + 12 q^{89}+O(q^{100}) \) 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
−0.618034
1.61803
0 −1.23607 0 0 0 5.23607 0 −1.47214 0
1.2 0 3.23607 0 0 0 0.763932 0 7.47214 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.a.n 2
3.b odd 2 1 7200.2.a.cr 2
4.b odd 2 1 800.2.a.j 2
5.b even 2 1 800.2.a.j 2
5.c odd 4 2 160.2.c.b 4
8.b even 2 1 1600.2.a.z 2
8.d odd 2 1 1600.2.a.bd 2
12.b even 2 1 7200.2.a.cb 2
15.d odd 2 1 7200.2.a.cb 2
15.e even 4 2 1440.2.f.i 4
20.d odd 2 1 CM 800.2.a.n 2
20.e even 4 2 160.2.c.b 4
40.e odd 2 1 1600.2.a.z 2
40.f even 2 1 1600.2.a.bd 2
40.i odd 4 2 320.2.c.d 4
40.k even 4 2 320.2.c.d 4
60.h even 2 1 7200.2.a.cr 2
60.l odd 4 2 1440.2.f.i 4
80.i odd 4 2 1280.2.f.g 4
80.j even 4 2 1280.2.f.g 4
80.s even 4 2 1280.2.f.h 4
80.t odd 4 2 1280.2.f.h 4
120.q odd 4 2 2880.2.f.w 4
120.w even 4 2 2880.2.f.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 5.c odd 4 2
160.2.c.b 4 20.e even 4 2
320.2.c.d 4 40.i odd 4 2
320.2.c.d 4 40.k even 4 2
800.2.a.j 2 4.b odd 2 1
800.2.a.j 2 5.b even 2 1
800.2.a.n 2 1.a even 1 1 trivial
800.2.a.n 2 20.d odd 2 1 CM
1280.2.f.g 4 80.i odd 4 2
1280.2.f.g 4 80.j even 4 2
1280.2.f.h 4 80.s even 4 2
1280.2.f.h 4 80.t odd 4 2
1440.2.f.i 4 15.e even 4 2
1440.2.f.i 4 60.l odd 4 2
1600.2.a.z 2 8.b even 2 1
1600.2.a.z 2 40.e odd 2 1
1600.2.a.bd 2 8.d odd 2 1
1600.2.a.bd 2 40.f even 2 1
2880.2.f.w 4 120.q odd 4 2
2880.2.f.w 4 120.w even 4 2
7200.2.a.cb 2 12.b even 2 1
7200.2.a.cb 2 15.d odd 2 1
7200.2.a.cr 2 3.b odd 2 1
7200.2.a.cr 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(800))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 20 \) Copy content Toggle raw display
$43$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 180 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 116 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 22T + 76 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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