Properties

Label 800.4.a.l
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $1$
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] = [800,4,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: \(2\)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{3} + ( - 11 \beta - 9) q^{7} + (14 \beta + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{3} + ( - 11 \beta - 9) q^{7} + (14 \beta + 27) q^{9} + (86 \beta + 118) q^{21} + (63 \beta - 67) q^{23} + ( - 98 \beta - 70) q^{27} + 306 q^{29} - 206 \beta q^{41} + (119 \beta + 297) q^{43} + (153 \beta - 301) q^{47} + (198 \beta + 343) q^{49} - 18 \beta q^{61} + ( - 423 \beta - 1013) q^{63} + (245 \beta - 549) q^{67} + ( - 374 \beta + 154) q^{69} + (378 \beta + 251) q^{81} + ( - 477 \beta + 77) q^{83} + ( - 306 \beta - 2142) q^{87} - 1386 q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 18 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} - 18 q^{7} + 54 q^{9} + 236 q^{21} - 134 q^{23} - 140 q^{27} + 612 q^{29} + 594 q^{43} - 602 q^{47} + 686 q^{49} - 2026 q^{63} - 1098 q^{67} + 308 q^{69} + 502 q^{81} + 154 q^{83} - 4284 q^{87} - 2772 q^{89}+O(q^{100}) \) 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
1.61803
−0.618034
0 −9.23607 0 0 0 −33.5967 0 58.3050 0
1.2 0 −4.76393 0 0 0 15.5967 0 −4.30495 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 800.4.a.l 2
4.b odd 2 1 800.4.a.t 2
5.b even 2 1 800.4.a.t 2
5.c odd 4 2 160.4.c.c 4
8.b even 2 1 1600.4.a.cp 2
8.d odd 2 1 1600.4.a.cb 2
15.e even 4 2 1440.4.f.g 4
20.d odd 2 1 CM 800.4.a.l 2
20.e even 4 2 160.4.c.c 4
40.e odd 2 1 1600.4.a.cp 2
40.f even 2 1 1600.4.a.cb 2
40.i odd 4 2 320.4.c.f 4
40.k even 4 2 320.4.c.f 4
60.l odd 4 2 1440.4.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.c 4 5.c odd 4 2
160.4.c.c 4 20.e even 4 2
320.4.c.f 4 40.i odd 4 2
320.4.c.f 4 40.k even 4 2
800.4.a.l 2 1.a even 1 1 trivial
800.4.a.l 2 20.d odd 2 1 CM
800.4.a.t 2 4.b odd 2 1
800.4.a.t 2 5.b even 2 1
1440.4.f.g 4 15.e even 4 2
1440.4.f.g 4 60.l odd 4 2
1600.4.a.cb 2 8.d odd 2 1
1600.4.a.cb 2 40.f even 2 1
1600.4.a.cp 2 8.b even 2 1
1600.4.a.cp 2 40.e odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 14T_{3} + 44 \) 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
$3$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 18T - 524 \) 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
$23$ \( T^{2} + 134T - 15356 \) Copy content Toggle raw display
$29$ \( (T - 306)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 212180 \) Copy content Toggle raw display
$43$ \( T^{2} - 594T + 17404 \) Copy content Toggle raw display
$47$ \( T^{2} + 602T - 26444 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 1620 \) Copy content Toggle raw display
$67$ \( T^{2} + 1098T + 1276 \) 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
$83$ \( T^{2} - 154 T - 1131716 \) Copy content Toggle raw display
$89$ \( (T + 1386)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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