↑

Properties

Label 560.2.ci.e
Level $560$
Weight $2$
Character orbit 560.ci
Analytic conductor $4.472$
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] = [560,2,Mod(17,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.ci (of order \(12\), degree \(4\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 280)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{7} - 4 q^{11} - 8 q^{15} - 4 q^{21} + 4 q^{23} - 8 q^{25} - 36 q^{33} - 24 q^{35} + 8 q^{37} + 16 q^{43} + 48 q^{45} - 24 q^{51} + 16 q^{53} - 96 q^{57} - 36 q^{61} + 68 q^{63} + 12 q^{65} + 16 q^{67} + 64 q^{71} - 48 q^{73} + 48 q^{75} + 4 q^{77} - 40 q^{85} + 12 q^{87} + 80 q^{91} - 24 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −3.19510 + 0.856125i 0 0.672461 + 2.13256i 0 2.52104 + 0.802724i 0 6.87765 − 3.97081i 0
17.2 0 −2.20364 + 0.590462i 0 −0.352053 − 2.20818i 0 1.22435 + 2.34541i 0 1.90929 − 1.10233i 0
17.3 0 −2.19810 + 0.588978i 0 −1.19880 − 1.88756i 0 −1.40409 − 2.24244i 0 1.88665 − 1.08926i 0
17.4 0 −1.66856 + 0.447090i 0 2.21600 + 0.298941i 0 −0.900705 − 2.48772i 0 −0.0138674 + 0.00800633i 0
17.5 0 −0.280752 + 0.0752271i 0 2.21121 − 0.332503i 0 −1.13104 + 2.39181i 0 −2.52491 + 1.45776i 0
17.6 0 −0.120281 + 0.0322291i 0 −2.04377 + 0.907205i 0 2.46167 − 0.969635i 0 −2.58465 + 1.49225i 0
17.7 0 0.0749389 − 0.0200798i 0 −0.965007 + 2.01712i 0 −1.45892 − 2.20716i 0 −2.59286 + 1.49699i 0
17.8 0 1.16710 − 0.312724i 0 −0.121837 − 2.23275i 0 2.59977 − 0.491095i 0 −1.33374 + 0.770036i 0
17.9 0 1.35944 − 0.364260i 0 −2.13431 − 0.666887i 0 −2.59782 + 0.501338i 0 −0.882692 + 0.509622i 0
17.10 0 2.00047 − 0.536025i 0 1.38383 + 1.75642i 0 −1.24986 + 2.33192i 0 1.11649 − 0.644605i 0
17.11 0 2.30560 − 0.617784i 0 1.37859 − 1.76054i 0 0.755351 − 2.53563i 0 2.33607 − 1.34873i 0
17.12 0 2.75887 − 0.739238i 0 −1.04631 + 1.97617i 0 1.91229 + 1.82843i 0 4.46683 − 2.57893i 0
33.1 0 −3.19510 − 0.856125i 0 0.672461 − 2.13256i 0 2.52104 − 0.802724i 0 6.87765 + 3.97081i 0
33.2 0 −2.20364 − 0.590462i 0 −0.352053 + 2.20818i 0 1.22435 − 2.34541i 0 1.90929 + 1.10233i 0
33.3 0 −2.19810 − 0.588978i 0 −1.19880 + 1.88756i 0 −1.40409 + 2.24244i 0 1.88665 + 1.08926i 0
33.4 0 −1.66856 − 0.447090i 0 2.21600 − 0.298941i 0 −0.900705 + 2.48772i 0 −0.0138674 − 0.00800633i 0
33.5 0 −0.280752 − 0.0752271i 0 2.21121 + 0.332503i 0 −1.13104 − 2.39181i 0 −2.52491 − 1.45776i 0
33.6 0 −0.120281 − 0.0322291i 0 −2.04377 − 0.907205i 0 2.46167 + 0.969635i 0 −2.58465 − 1.49225i 0
33.7 0 0.0749389 + 0.0200798i 0 −0.965007 − 2.01712i 0 −1.45892 + 2.20716i 0 −2.59286 − 1.49699i 0
33.8 0 1.16710 + 0.312724i 0 −0.121837 + 2.23275i 0 2.59977 + 0.491095i 0 −1.33374 − 0.770036i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.ci.e 48
4.b odd 2 1 280.2.bo.a ✓ 48
5.c odd 4 1 inner 560.2.ci.e 48
7.d odd 6 1 inner 560.2.ci.e 48
20.e even 4 1 280.2.bo.a ✓ 48
28.f even 6 1 280.2.bo.a ✓ 48
35.k even 12 1 inner 560.2.ci.e 48
140.x odd 12 1 280.2.bo.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bo.a ✓ 48 4.b odd 2 1
280.2.bo.a ✓ 48 20.e even 4 1
280.2.bo.a ✓ 48 28.f even 6 1
280.2.bo.a ✓ 48 140.x odd 12 1
560.2.ci.e 48 1.a even 1 1 trivial
560.2.ci.e 48 5.c odd 4 1 inner
560.2.ci.e 48 7.d odd 6 1 inner
560.2.ci.e 48 35.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 153 T_{3}^{44} + 108 T_{3}^{43} - 996 T_{3}^{41} + 17131 T_{3}^{40} - 16524 T_{3}^{39} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display