↑

Properties

Label 640.4.l.a
Level $640$
Weight $4$
Character orbit 640.l
Analytic conductor $37.761$
Analytic rank $0$
Dimension $48$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,4,Mod(161,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.161");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 640.l (of order \(4\), degree \(2\), 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: \(37.7612224037\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 264 q^{27} - 400 q^{29} - 16 q^{37} + 808 q^{43} - 1880 q^{47} - 2352 q^{49} + 2144 q^{51} - 752 q^{53} - 2728 q^{59} + 912 q^{61} + 2520 q^{63} - 2040 q^{67}+ \cdots - 4456 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 −6.54282 − 6.54282i 0 3.53553 − 3.53553i 0 − 11.7588i 0 58.6170i 0
161.2 0 −6.39295 − 6.39295i 0 3.53553 − 3.53553i 0 30.7844i 0 54.7397i 0
161.3 0 −5.98518 − 5.98518i 0 −3.53553 + 3.53553i 0 − 30.6332i 0 44.6447i 0
161.4 0 −4.91823 − 4.91823i 0 −3.53553 + 3.53553i 0 29.5330i 0 21.3780i 0
161.5 0 −4.76629 − 4.76629i 0 3.53553 − 3.53553i 0 − 25.1089i 0 18.4350i 0
161.6 0 −4.63181 − 4.63181i 0 −3.53553 + 3.53553i 0 8.38903i 0 15.9074i 0
161.7 0 −2.72881 − 2.72881i 0 −3.53553 + 3.53553i 0 − 9.20767i 0 − 12.1072i 0
161.8 0 −2.65806 − 2.65806i 0 3.53553 − 3.53553i 0 − 21.5840i 0 − 12.8694i 0
161.9 0 −2.10736 − 2.10736i 0 −3.53553 + 3.53553i 0 − 1.52539i 0 − 18.1181i 0
161.10 0 −1.80275 − 1.80275i 0 3.53553 − 3.53553i 0 5.36354i 0 − 20.5002i 0
161.11 0 −1.79271 − 1.79271i 0 3.53553 − 3.53553i 0 − 14.5354i 0 − 20.5724i 0
161.12 0 0.0794032 + 0.0794032i 0 3.53553 − 3.53553i 0 17.2069i 0 − 26.9874i 0
161.13 0 0.332572 + 0.332572i 0 −3.53553 + 3.53553i 0 − 31.6472i 0 − 26.7788i 0
161.14 0 0.777426 + 0.777426i 0 3.53553 − 3.53553i 0 32.6046i 0 − 25.7912i 0
161.15 0 1.30312 + 1.30312i 0 −3.53553 + 3.53553i 0 17.9744i 0 − 23.6037i 0
161.16 0 1.69929 + 1.69929i 0 −3.53553 + 3.53553i 0 12.7537i 0 − 21.2249i 0
161.17 0 3.85223 + 3.85223i 0 3.53553 − 3.53553i 0 − 22.7583i 0 2.67940i 0
161.18 0 3.98819 + 3.98819i 0 −3.53553 + 3.53553i 0 − 23.3105i 0 4.81140i 0
161.19 0 4.03507 + 4.03507i 0 3.53553 − 3.53553i 0 2.50273i 0 5.56352i 0
161.20 0 4.03778 + 4.03778i 0 3.53553 − 3.53553i 0 − 8.24125i 0 5.60734i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.4.l.a 48
4.b odd 2 1 640.4.l.b 48
8.b even 2 1 320.4.l.a 48
8.d odd 2 1 80.4.l.a ✓ 48
16.e even 4 1 320.4.l.a 48
16.e even 4 1 inner 640.4.l.a 48
16.f odd 4 1 80.4.l.a ✓ 48
16.f odd 4 1 640.4.l.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.4.l.a ✓ 48 8.d odd 2 1
80.4.l.a ✓ 48 16.f odd 4 1
320.4.l.a 48 8.b even 2 1
320.4.l.a 48 16.e even 4 1
640.4.l.a 48 1.a even 1 1 trivial
640.4.l.a 48 16.e even 4 1 inner
640.4.l.b 48 4.b odd 2 1
640.4.l.b 48 16.f odd 4 1