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Properties

Label 640.3.h.a
Level $640$
Weight $3$
Character orbit 640.h
Analytic conductor $17.439$
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] = [640,3,Mod(639,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.639");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 640.h (of order \(2\), degree \(1\), 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: \(17.4387369191\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{5} + 72 q^{9} - 32 q^{21} + 24 q^{25} - 16 q^{29} + 80 q^{41} - 120 q^{45} + 168 q^{49} - 400 q^{61} + 32 q^{65} - 224 q^{69} + 184 q^{81} - 288 q^{85} + 144 q^{89}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
639.1 0 −5.46407 0 −3.35247 − 3.70957i 0 −5.17181 0 20.8560 0
639.2 0 −5.46407 0 −3.35247 + 3.70957i 0 −5.17181 0 20.8560 0
639.3 0 −4.20423 0 4.76298 − 1.52118i 0 −3.29905 0 8.67551 0
639.4 0 −4.20423 0 4.76298 + 1.52118i 0 −3.29905 0 8.67551 0
639.5 0 −4.06912 0 −2.41379 − 4.37877i 0 13.1670 0 7.55776 0
639.6 0 −4.06912 0 −2.41379 + 4.37877i 0 13.1670 0 7.55776 0
639.7 0 −2.00470 0 0.247851 − 4.99385i 0 −4.85433 0 −4.98119 0
639.8 0 −2.00470 0 0.247851 + 4.99385i 0 −4.85433 0 −4.98119 0
639.9 0 −1.73799 0 3.70279 − 3.35996i 0 7.32638 0 −5.97938 0
639.10 0 −1.73799 0 3.70279 + 3.35996i 0 7.32638 0 −5.97938 0
639.11 0 −0.933412 0 −4.94736 + 0.723593i 0 −6.91082 0 −8.12874 0
639.12 0 −0.933412 0 −4.94736 − 0.723593i 0 −6.91082 0 −8.12874 0
639.13 0 0.933412 0 −4.94736 − 0.723593i 0 6.91082 0 −8.12874 0
639.14 0 0.933412 0 −4.94736 + 0.723593i 0 6.91082 0 −8.12874 0
639.15 0 1.73799 0 3.70279 + 3.35996i 0 −7.32638 0 −5.97938 0
639.16 0 1.73799 0 3.70279 − 3.35996i 0 −7.32638 0 −5.97938 0
639.17 0 2.00470 0 0.247851 + 4.99385i 0 4.85433 0 −4.98119 0
639.18 0 2.00470 0 0.247851 − 4.99385i 0 4.85433 0 −4.98119 0
639.19 0 4.06912 0 −2.41379 + 4.37877i 0 −13.1670 0 7.55776 0
639.20 0 4.06912 0 −2.41379 − 4.37877i 0 −13.1670 0 7.55776 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 639.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.3.h.a ✓ 24
4.b odd 2 1 inner 640.3.h.a ✓ 24
5.b even 2 1 inner 640.3.h.a ✓ 24
8.b even 2 1 640.3.h.b yes 24
8.d odd 2 1 640.3.h.b yes 24
16.e even 4 1 1280.3.e.k 24
16.e even 4 1 1280.3.e.l 24
16.f odd 4 1 1280.3.e.k 24
16.f odd 4 1 1280.3.e.l 24
20.d odd 2 1 inner 640.3.h.a ✓ 24
40.e odd 2 1 640.3.h.b yes 24
40.f even 2 1 640.3.h.b yes 24
80.k odd 4 1 1280.3.e.k 24
80.k odd 4 1 1280.3.e.l 24
80.q even 4 1 1280.3.e.k 24
80.q even 4 1 1280.3.e.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.3.h.a ✓ 24 1.a even 1 1 trivial
640.3.h.a ✓ 24 4.b odd 2 1 inner
640.3.h.a ✓ 24 5.b even 2 1 inner
640.3.h.a ✓ 24 20.d odd 2 1 inner
640.3.h.b yes 24 8.b even 2 1
640.3.h.b yes 24 8.d odd 2 1
640.3.h.b yes 24 40.e odd 2 1
640.3.h.b yes 24 40.f even 2 1
1280.3.e.k 24 16.e even 4 1
1280.3.e.k 24 16.f odd 4 1
1280.3.e.k 24 80.k odd 4 1
1280.3.e.k 24 80.q even 4 1
1280.3.e.l 24 16.e even 4 1
1280.3.e.l 24 16.f odd 4 1
1280.3.e.l 24 80.k odd 4 1
1280.3.e.l 24 80.q even 4 1