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Properties

Label 800.4.f.d
Level $800$
Weight $4$
Character orbit 800.f
Analytic conductor $47.202$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(49,800)]
 
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, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 216 q^{9} + 528 q^{31} - 600 q^{39} - 40 q^{41} - 456 q^{49} - 1256 q^{71} + 2232 q^{79} + 2256 q^{81} + 848 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −9.63387 0 0 0 21.1900i 0 65.8115 0
49.2 0 −9.63387 0 0 0 − 21.1900i 0 65.8115 0
49.3 0 −7.69300 0 0 0 − 15.6248i 0 32.1823 0
49.4 0 −7.69300 0 0 0 15.6248i 0 32.1823 0
49.5 0 −5.31349 0 0 0 − 15.7169i 0 1.23319 0
49.6 0 −5.31349 0 0 0 15.7169i 0 1.23319 0
49.7 0 −5.16961 0 0 0 − 7.07059i 0 −0.275157 0
49.8 0 −5.16961 0 0 0 7.07059i 0 −0.275157 0
49.9 0 −2.73090 0 0 0 31.2997i 0 −19.5422 0
49.10 0 −2.73090 0 0 0 − 31.2997i 0 −19.5422 0
49.11 0 −1.26108 0 0 0 − 14.2186i 0 −25.4097 0
49.12 0 −1.26108 0 0 0 14.2186i 0 −25.4097 0
49.13 0 1.26108 0 0 0 − 14.2186i 0 −25.4097 0
49.14 0 1.26108 0 0 0 14.2186i 0 −25.4097 0
49.15 0 2.73090 0 0 0 31.2997i 0 −19.5422 0
49.16 0 2.73090 0 0 0 − 31.2997i 0 −19.5422 0
49.17 0 5.16961 0 0 0 − 7.07059i 0 −0.275157 0
49.18 0 5.16961 0 0 0 7.07059i 0 −0.275157 0
49.19 0 5.31349 0 0 0 − 15.7169i 0 1.23319 0
49.20 0 5.31349 0 0 0 15.7169i 0 1.23319 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.f.d 24
4.b odd 2 1 200.4.f.d 24
5.b even 2 1 inner 800.4.f.d 24
5.c odd 4 1 800.4.d.b 12
5.c odd 4 1 800.4.d.c 12
8.b even 2 1 inner 800.4.f.d 24
8.d odd 2 1 200.4.f.d 24
20.d odd 2 1 200.4.f.d 24
20.e even 4 1 200.4.d.c ✓ 12
20.e even 4 1 200.4.d.d yes 12
40.e odd 2 1 200.4.f.d 24
40.f even 2 1 inner 800.4.f.d 24
40.i odd 4 1 800.4.d.b 12
40.i odd 4 1 800.4.d.c 12
40.k even 4 1 200.4.d.c ✓ 12
40.k even 4 1 200.4.d.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.d.c ✓ 12 20.e even 4 1
200.4.d.c ✓ 12 40.k even 4 1
200.4.d.d yes 12 20.e even 4 1
200.4.d.d yes 12 40.k even 4 1
200.4.f.d 24 4.b odd 2 1
200.4.f.d 24 8.d odd 2 1
200.4.f.d 24 20.d odd 2 1
200.4.f.d 24 40.e odd 2 1
800.4.d.b 12 5.c odd 4 1
800.4.d.b 12 40.i odd 4 1
800.4.d.c 12 5.c odd 4 1
800.4.d.c 12 40.i odd 4 1
800.4.f.d 24 1.a even 1 1 trivial
800.4.f.d 24 5.b even 2 1 inner
800.4.f.d 24 8.b even 2 1 inner
800.4.f.d 24 40.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 216T_{3}^{10} + 16485T_{3}^{8} - 551120T_{3}^{6} + 8086707T_{3}^{4} - 42440280T_{3}^{2} + 49154791 \) acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display