↑

Properties

Label 720.2.z.i
Level $720$
Weight $2$
Character orbit 720.z
Analytic conductor $5.749$
Analytic rank $0$
Dimension $48$
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] = [720,2,Mod(163,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.z (of order \(4\), degree \(2\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4} + 4 q^{16} + 8 q^{19} + 12 q^{22} - 4 q^{28} + 12 q^{34} + 28 q^{40} - 4 q^{46} - 64 q^{52} - 12 q^{58} - 16 q^{61} - 4 q^{64} - 84 q^{70} - 16 q^{73} - 36 q^{76} - 80 q^{82} - 60 q^{88} + 32 q^{91} + 28 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −1.39332 + 0.242199i 0 1.88268 − 0.674922i 1.62871 − 1.53209i 0 −3.13217 + 3.13217i −2.45971 + 1.39637i 0 −1.89825 + 2.52916i
163.2 −1.39270 + 0.245720i 0 1.87924 − 0.684429i −0.0729095 + 2.23488i 0 0.727248 − 0.727248i −2.44905 + 1.41497i 0 −0.447612 − 3.13044i
163.3 −1.36281 − 0.377823i 0 1.71450 + 1.02980i −0.928262 − 2.03429i 0 2.37620 − 2.37620i −1.94746 − 2.05120i 0 0.496444 + 3.12307i
163.4 −1.22436 + 0.707771i 0 0.998120 − 1.73313i −2.22028 − 0.265256i 0 −0.602536 + 0.602536i 0.00460245 + 2.82842i 0 2.90616 − 1.24668i
163.5 −1.16125 − 0.807156i 0 0.696998 + 1.87462i −1.83722 + 1.27461i 0 −0.821218 + 0.821218i 0.703721 − 2.73948i 0 3.16228 + 0.00278061i
163.6 −0.959371 + 1.03904i 0 −0.159215 − 1.99365i 2.20881 − 0.348049i 0 1.71419 − 1.71419i 2.22423 + 1.74722i 0 −1.75743 + 2.62896i
163.7 −0.909838 − 1.08268i 0 −0.344388 + 1.97013i 1.20506 − 1.88357i 0 −1.79235 + 1.79235i 2.44635 − 1.41963i 0 −3.13571 + 0.409050i
163.8 −0.723808 − 1.21495i 0 −0.952204 + 1.75878i 2.18764 + 0.462835i 0 3.51066 − 3.51066i 2.82604 − 0.116140i 0 −1.02111 − 2.99288i
163.9 −0.717796 + 1.21851i 0 −0.969539 − 1.74928i −0.913778 − 2.04084i 0 −0.371822 + 0.371822i 2.82745 + 0.0742344i 0 3.14269 + 0.351454i
163.10 −0.262580 + 1.38962i 0 −1.86210 − 0.729774i −0.707148 + 2.12131i 0 2.15752 − 2.15752i 1.50306 − 2.39600i 0 −2.76213 − 1.53968i
163.11 −0.203497 − 1.39950i 0 −1.91718 + 0.569587i 1.24936 + 1.85448i 0 −2.43860 + 2.43860i 1.18728 + 2.56717i 0 2.34110 − 2.12586i
163.12 −0.128620 − 1.40835i 0 −1.96691 + 0.362283i −2.04023 − 0.915132i 0 −1.32711 + 1.32711i 0.763206 + 2.72351i 0 −1.02641 + 2.99107i
163.13 0.128620 + 1.40835i 0 −1.96691 + 0.362283i 2.04023 + 0.915132i 0 −1.32711 + 1.32711i −0.763206 − 2.72351i 0 −1.02641 + 2.99107i
163.14 0.203497 + 1.39950i 0 −1.91718 + 0.569587i −1.24936 − 1.85448i 0 −2.43860 + 2.43860i −1.18728 − 2.56717i 0 2.34110 − 2.12586i
163.15 0.262580 − 1.38962i 0 −1.86210 − 0.729774i 0.707148 − 2.12131i 0 2.15752 − 2.15752i −1.50306 + 2.39600i 0 −2.76213 − 1.53968i
163.16 0.717796 − 1.21851i 0 −0.969539 − 1.74928i 0.913778 + 2.04084i 0 −0.371822 + 0.371822i −2.82745 − 0.0742344i 0 3.14269 + 0.351454i
163.17 0.723808 + 1.21495i 0 −0.952204 + 1.75878i −2.18764 − 0.462835i 0 3.51066 − 3.51066i −2.82604 + 0.116140i 0 −1.02111 − 2.99288i
163.18 0.909838 + 1.08268i 0 −0.344388 + 1.97013i −1.20506 + 1.88357i 0 −1.79235 + 1.79235i −2.44635 + 1.41963i 0 −3.13571 + 0.409050i
163.19 0.959371 − 1.03904i 0 −0.159215 − 1.99365i −2.20881 + 0.348049i 0 1.71419 − 1.71419i −2.22423 − 1.74722i 0 −1.75743 + 2.62896i
163.20 1.16125 + 0.807156i 0 0.696998 + 1.87462i 1.83722 − 1.27461i 0 −0.821218 + 0.821218i −0.703721 + 2.73948i 0 3.16228 + 0.00278061i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
80.s even 4 1 inner
240.z odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.z.i ✓ 48
3.b odd 2 1 inner 720.2.z.i ✓ 48
5.c odd 4 1 720.2.bd.i yes 48
15.e even 4 1 720.2.bd.i yes 48
16.f odd 4 1 720.2.bd.i yes 48
48.k even 4 1 720.2.bd.i yes 48
80.s even 4 1 inner 720.2.z.i ✓ 48
240.z odd 4 1 inner 720.2.z.i ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.z.i ✓ 48 1.a even 1 1 trivial
720.2.z.i ✓ 48 3.b odd 2 1 inner
720.2.z.i ✓ 48 80.s even 4 1 inner
720.2.z.i ✓ 48 240.z odd 4 1 inner
720.2.bd.i yes 48 5.c odd 4 1
720.2.bd.i yes 48 15.e even 4 1
720.2.bd.i yes 48 16.f odd 4 1
720.2.bd.i yes 48 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{24} + 24 T_{7}^{21} + 720 T_{7}^{20} + 416 T_{7}^{19} + 288 T_{7}^{18} + 8768 T_{7}^{17} + \cdots + 23040000 \) Copy content Toggle raw display
\( T_{11}^{48} + 3328 T_{11}^{44} + 3734144 T_{11}^{40} + 2049663232 T_{11}^{36} + 619735267328 T_{11}^{32} + \cdots + 875781160960000 \) Copy content Toggle raw display