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Properties

Label 320.4.j.a
Level $320$
Weight $4$
Character orbit 320.j
Analytic conductor $18.881$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(47,320)]
 
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([2, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.j (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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{5} + 4 q^{7} - 540 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 2 q^{5} + 4 q^{7} - 540 q^{9} + 4 q^{11} - 4 q^{13} - 108 q^{15} - 4 q^{17} + 24 q^{19} - 4 q^{21} + 4 q^{23} - 4 q^{33} + 480 q^{35} - 4 q^{37} - 860 q^{43} - 198 q^{45} + 408 q^{47} - 740 q^{51} + 4 q^{55} + 108 q^{57} - 688 q^{59} - 916 q^{61} - 108 q^{63} - 4 q^{65} - 1844 q^{67} - 420 q^{69} + 232 q^{71} - 296 q^{73} + 496 q^{75} + 2908 q^{81} - 252 q^{85} + 1292 q^{87} - 844 q^{91} + 104 q^{93} - 1240 q^{95} - 4 q^{97} - 2764 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 − 9.95697i 0 9.96550 − 5.06841i 0 9.89810 + 9.89810i 0 −72.1412 0
47.2 0 − 9.57547i 0 −0.434283 + 11.1719i 0 8.07205 + 8.07205i 0 −64.6895 0
47.3 0 − 9.06020i 0 −6.52613 − 9.07797i 0 −20.8390 − 20.8390i 0 −55.0872 0
47.4 0 − 7.89491i 0 −9.18689 − 6.37189i 0 10.3963 + 10.3963i 0 −35.3295 0
47.5 0 − 6.67181i 0 7.80689 + 8.00328i 0 −20.5642 − 20.5642i 0 −17.5131 0
47.6 0 − 6.56575i 0 −7.38475 + 8.39437i 0 −18.3319 − 18.3319i 0 −16.1090 0
47.7 0 − 5.84366i 0 7.49235 − 8.29848i 0 −4.15713 − 4.15713i 0 −7.14836 0
47.8 0 − 5.75615i 0 0.458746 − 11.1709i 0 20.4957 + 20.4957i 0 −6.13323 0
47.9 0 − 5.41494i 0 −5.16149 + 9.91761i 0 17.1720 + 17.1720i 0 −2.32163 0
47.10 0 − 4.83638i 0 −10.5128 + 3.80550i 0 6.67315 + 6.67315i 0 3.60946 0
47.11 0 − 4.66716i 0 10.0057 + 4.98857i 0 3.20255 + 3.20255i 0 5.21762 0
47.12 0 − 4.64797i 0 10.0868 + 4.82255i 0 −1.85535 − 1.85535i 0 5.39640 0
47.13 0 − 2.90856i 0 −6.79776 − 8.87640i 0 −7.30822 − 7.30822i 0 18.5403 0
47.14 0 − 2.49888i 0 6.71701 − 8.93766i 0 −7.69858 − 7.69858i 0 20.7556 0
47.15 0 − 2.08304i 0 −11.1381 + 0.971164i 0 −1.79251 − 1.79251i 0 22.6610 0
47.16 0 0.100896i 0 0.997085 − 11.1358i 0 −16.7722 − 16.7722i 0 26.9898 0
47.17 0 0.596808i 0 −6.90677 − 8.79185i 0 19.7006 + 19.7006i 0 26.6438 0
47.18 0 0.816587i 0 −3.81941 + 10.5077i 0 −14.6998 − 14.6998i 0 26.3332 0
47.19 0 0.998127i 0 10.9800 − 2.10693i 0 25.3753 + 25.3753i 0 26.0037 0
47.20 0 1.42375i 0 5.36673 + 9.80807i 0 3.65126 + 3.65126i 0 24.9729 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j 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.j.a 68
4.b odd 2 1 80.4.j.a ✓ 68
5.c odd 4 1 320.4.s.a 68
16.e even 4 1 80.4.s.a yes 68
16.f odd 4 1 320.4.s.a 68
20.e even 4 1 80.4.s.a yes 68
80.j even 4 1 inner 320.4.j.a 68
80.t odd 4 1 80.4.j.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.4.j.a ✓ 68 4.b odd 2 1
80.4.j.a ✓ 68 80.t odd 4 1
80.4.s.a yes 68 16.e even 4 1
80.4.s.a yes 68 20.e even 4 1
320.4.j.a 68 1.a even 1 1 trivial
320.4.j.a 68 80.j even 4 1 inner
320.4.s.a 68 5.c odd 4 1
320.4.s.a 68 16.f odd 4 1

Hecke kernels

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