↑

Properties

Label 520.2.cu.a
Level $520$
Weight $2$
Character orbit 520.cu
Analytic conductor $4.152$
Analytic rank $0$
Dimension $320$
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] = [520,2,Mod(197,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.cu (of order \(12\), degree \(4\), 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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(80\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{2} - 8 q^{6} - 12 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{2} - 8 q^{6} - 12 q^{7} - 8 q^{8} - 2 q^{10} + 16 q^{12} - 32 q^{15} - 4 q^{16} - 8 q^{17} - 22 q^{20} - 4 q^{22} - 12 q^{23} + 8 q^{24} - 24 q^{26} - 6 q^{28} + 2 q^{30} - 16 q^{31} + 8 q^{32} - 28 q^{33} - 8 q^{34} - 12 q^{36} - 8 q^{38} - 64 q^{40} - 16 q^{41} - 80 q^{42} - 24 q^{44} - 8 q^{46} + 64 q^{48} + 112 q^{49} - 14 q^{52} - 48 q^{54} - 4 q^{55} - 12 q^{56} + 8 q^{57} + 66 q^{58} - 20 q^{60} + 8 q^{62} - 48 q^{63} - 36 q^{65} - 16 q^{66} - 12 q^{68} - 32 q^{70} + 16 q^{71} - 54 q^{72} - 56 q^{73} + 48 q^{74} - 40 q^{76} + 132 q^{78} + 42 q^{80} + 88 q^{81} + 10 q^{82} - 88 q^{84} - 12 q^{86} + 20 q^{87} - 94 q^{88} - 112 q^{90} + 8 q^{92} + 24 q^{94} - 12 q^{95} + 48 q^{96} - 12 q^{97} + 38 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.41410 + 0.0182725i 0.287350 − 1.07240i 1.99933 − 0.0516782i −2.23587 + 0.0295312i −0.386745 + 1.52173i 1.73120 + 0.999510i −2.82630 + 0.109611i 1.53059 + 0.883689i 3.16120 − 0.0826151i
197.2 −1.40707 + 0.141916i −0.645100 + 2.40755i 1.95972 − 0.399374i 0.616611 + 2.14937i 0.566034 − 3.47915i 0.195130 + 0.112658i −2.70079 + 0.840066i −2.78205 − 1.60622i −1.17265 − 2.93682i
197.3 −1.40567 − 0.155250i −0.540244 + 2.01622i 1.95180 + 0.436458i 0.716861 − 2.11804i 1.07242 − 2.75026i −3.69977 − 2.13606i −2.67581 − 0.916530i −1.17519 − 0.678498i −1.33649 + 2.86597i
197.4 −1.40462 + 0.164476i −0.0665329 + 0.248304i 1.94589 − 0.462053i −1.81289 + 1.30897i 0.0526130 − 0.359715i −2.58884 − 1.49467i −2.65724 + 0.969061i 2.54085 + 1.46696i 2.33113 − 2.13678i
197.5 −1.38942 + 0.263636i −0.0426396 + 0.159133i 1.86099 − 0.732604i 1.43369 − 1.71596i 0.0172912 − 0.232344i 4.06938 + 2.34945i −2.39256 + 1.50852i 2.57457 + 1.48643i −1.53961 + 2.76217i
197.6 −1.38651 + 0.278552i 0.698764 − 2.60782i 1.84482 − 0.772430i 1.40563 − 1.73903i −0.242429 + 3.81041i −1.15224 − 0.665245i −2.34270 + 1.58486i −3.71439 − 2.14450i −1.46451 + 2.80272i
197.7 −1.37995 − 0.309407i −0.401366 + 1.49792i 1.80853 + 0.853934i 1.97125 + 1.05554i 1.01733 − 1.94287i 0.900070 + 0.519656i −2.23148 − 1.73796i 0.515413 + 0.297574i −2.39365 − 2.06651i
197.8 −1.36451 − 0.371621i 0.630173 − 2.35184i 1.72380 + 1.01416i 1.56951 + 1.59269i −1.73387 + 2.97493i 2.75830 + 1.59251i −1.97526 − 2.02444i −2.53594 − 1.46413i −1.54974 − 2.75650i
197.9 −1.34637 + 0.432761i −0.825427 + 3.08054i 1.62544 − 1.16532i −1.43413 − 1.71559i −0.221805 − 4.50476i 1.98510 + 1.14610i −1.68414 + 2.27237i −6.21030 − 3.58552i 2.67332 + 1.68919i
197.10 −1.31256 − 0.526491i 0.863512 − 3.22267i 1.44562 + 1.38210i −2.05399 + 0.883806i −2.83012 + 3.77531i −2.84699 − 1.64371i −1.16979 − 2.57519i −7.04188 − 4.06563i 3.16130 − 0.0786389i
197.11 −1.29824 − 0.560857i 0.287680 − 1.07364i 1.37088 + 1.45626i 2.22712 − 0.199881i −0.975636 + 1.23250i −2.91113 − 1.68074i −0.962986 − 2.65945i 1.52814 + 0.882271i −3.00345 − 0.989598i
197.12 −1.29525 − 0.567747i 0.117735 − 0.439392i 1.35533 + 1.47074i −0.791031 − 2.09148i −0.401959 + 0.502277i −1.01386 − 0.585353i −0.920474 − 2.67446i 2.41887 + 1.39654i −0.162848 + 3.15808i
197.13 −1.27628 + 0.609178i 0.359784 − 1.34273i 1.25780 − 1.55497i 1.57663 + 1.58564i 0.358776 + 1.93288i −2.41190 − 1.39251i −0.658061 + 2.75081i 0.924595 + 0.533815i −2.97816 − 1.06328i
197.14 −1.20329 − 0.743028i 0.00756508 − 0.0282333i 0.895819 + 1.78816i −0.549598 + 2.16747i −0.0300811 + 0.0283518i 2.15884 + 1.24641i 0.250720 − 2.81729i 2.59734 + 1.49957i 2.27182 − 2.19974i
197.15 −1.18930 − 0.765217i −0.382990 + 1.42934i 0.828885 + 1.82015i −1.93908 − 1.11354i 1.54925 − 1.40685i 2.59477 + 1.49809i 0.407015 − 2.79899i 0.701748 + 0.405154i 1.45406 + 2.80815i
197.16 −1.18503 + 0.771815i 0.731736 − 2.73087i 0.808604 − 1.82925i −0.612584 + 2.15052i 1.24060 + 3.80094i 0.535091 + 0.308935i 0.453621 + 2.79181i −4.32416 − 2.49656i −0.933873 − 3.02124i
197.17 −1.18004 + 0.779431i −0.163881 + 0.611613i 0.784974 − 1.83951i −1.64149 − 1.51839i −0.283325 − 0.849461i −1.87761 − 1.08404i 0.507476 + 2.78253i 2.25086 + 1.29954i 3.12050 + 0.512326i
197.18 −1.15139 − 0.821161i −0.747776 + 2.79074i 0.651388 + 1.89095i −1.99268 + 1.01451i 3.15263 − 2.59918i −2.51629 − 1.45278i 0.802775 − 2.71211i −4.63097 − 2.67369i 3.12742 + 0.468224i
197.19 −1.11258 + 0.873023i −0.323028 + 1.20556i 0.475662 − 1.94261i −1.17801 + 1.90060i −0.693085 − 1.62329i 1.19182 + 0.688098i 1.16673 + 2.57657i 1.24905 + 0.721141i −0.348642 − 3.14300i
197.20 −0.992562 + 1.00738i 0.628474 − 2.34550i −0.0296401 − 1.99978i −1.99155 − 1.01672i 1.73901 + 2.96117i 4.04438 + 2.33503i 2.04396 + 1.95505i −2.50830 − 1.44817i 3.00097 − 0.997092i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
65.o even 12 1 inner
520.cu even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.cu.a yes 320
5.c odd 4 1 520.2.cj.a ✓ 320
8.b even 2 1 inner 520.2.cu.a yes 320
13.f odd 12 1 520.2.cj.a ✓ 320
40.i odd 4 1 520.2.cj.a ✓ 320
65.o even 12 1 inner 520.2.cu.a yes 320
104.x odd 12 1 520.2.cj.a ✓ 320
520.cu even 12 1 inner 520.2.cu.a yes 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.cj.a ✓ 320 5.c odd 4 1
520.2.cj.a ✓ 320 13.f odd 12 1
520.2.cj.a ✓ 320 40.i odd 4 1
520.2.cj.a ✓ 320 104.x odd 12 1
520.2.cu.a yes 320 1.a even 1 1 trivial
520.2.cu.a yes 320 8.b even 2 1 inner
520.2.cu.a yes 320 65.o even 12 1 inner
520.2.cu.a yes 320 520.cu even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(520, [\chi])\).