↑

Properties

Label 320.4.q.a
Level $320$
Weight $4$
Character orbit 320.q
Analytic conductor $18.881$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(49,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.q (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: \(18.8806112018\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{5} + 4 q^{11} + 4 q^{15} - 20 q^{19} - 112 q^{21} - 4 q^{29} + 752 q^{31} + 24 q^{35} + 302 q^{45} + 2148 q^{49} + 640 q^{51} + 692 q^{59} + 908 q^{61} + 484 q^{65} - 640 q^{69} + 604 q^{75} + 336 q^{79} - 2924 q^{81} + 248 q^{85} + 528 q^{91} - 3860 q^{95} - 364 q^{99}+O(q^{100}) \) 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 −7.10567 − 7.10567i 0 5.09839 − 9.95020i 0 −16.9895 0 73.9810i 0
49.2 0 −6.32652 − 6.32652i 0 −10.8777 − 2.58354i 0 27.8170 0 53.0497i 0
49.3 0 −6.32513 − 6.32513i 0 6.26783 + 9.25820i 0 17.8675 0 53.0146i 0
49.4 0 −5.25955 − 5.25955i 0 −6.37908 + 9.18190i 0 −2.79507 0 28.3257i 0
49.5 0 −5.16106 − 5.16106i 0 −0.347971 + 11.1749i 0 −22.6097 0 26.2731i 0
49.6 0 −4.90728 − 4.90728i 0 −8.53577 − 7.22084i 0 −16.1459 0 21.1628i 0
49.7 0 −4.74774 − 4.74774i 0 11.0045 + 1.97499i 0 −8.59244 0 18.0821i 0
49.8 0 −4.56462 − 4.56462i 0 2.80775 − 10.8220i 0 3.03301 0 14.6715i 0
49.9 0 −3.63859 − 3.63859i 0 10.7573 − 3.04646i 0 29.3525 0 − 0.521348i 0
49.10 0 −3.10771 − 3.10771i 0 −8.95059 + 6.69978i 0 22.2132 0 − 7.68424i 0
49.11 0 −2.67942 − 2.67942i 0 −11.1419 + 0.925976i 0 −35.2795 0 − 12.6414i 0
49.12 0 −2.09729 − 2.09729i 0 −8.07064 − 7.73723i 0 6.76050 0 − 18.2028i 0
49.13 0 −1.84963 − 1.84963i 0 −1.03217 − 11.1326i 0 17.3426 0 − 20.1577i 0
49.14 0 −1.64454 − 1.64454i 0 10.3044 + 4.33812i 0 11.1070 0 − 21.5909i 0
49.15 0 −1.28590 − 1.28590i 0 10.0922 + 4.81116i 0 −27.8938 0 − 23.6929i 0
49.16 0 −1.15578 − 1.15578i 0 5.16660 − 9.91495i 0 −15.7689 0 − 24.3283i 0
49.17 0 −1.06284 − 1.06284i 0 −3.67824 + 10.5580i 0 5.23459 0 − 24.7407i 0
49.18 0 1.06284 + 1.06284i 0 10.5580 − 3.67824i 0 −5.23459 0 − 24.7407i 0
49.19 0 1.15578 + 1.15578i 0 −9.91495 + 5.16660i 0 15.7689 0 − 24.3283i 0
49.20 0 1.28590 + 1.28590i 0 4.81116 + 10.0922i 0 27.8938 0 − 23.6929i 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.q.a 68
4.b odd 2 1 80.4.q.a ✓ 68
5.b even 2 1 inner 320.4.q.a 68
16.e even 4 1 inner 320.4.q.a 68
16.f odd 4 1 80.4.q.a ✓ 68
20.d odd 2 1 80.4.q.a ✓ 68
80.k odd 4 1 80.4.q.a ✓ 68
80.q even 4 1 inner 320.4.q.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.4.q.a ✓ 68 4.b odd 2 1
80.4.q.a ✓ 68 16.f odd 4 1
80.4.q.a ✓ 68 20.d odd 2 1
80.4.q.a ✓ 68 80.k odd 4 1
320.4.q.a 68 1.a even 1 1 trivial
320.4.q.a 68 5.b even 2 1 inner
320.4.q.a 68 16.e even 4 1 inner
320.4.q.a 68 80.q even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(320, [\chi])\).