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Properties

Label 585.2.p.b
Level $585$
Weight $2$
Character orbit 585.p
Analytic conductor $4.671$
Analytic rank $0$
Dimension $24$
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] = [585,2,Mod(53,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.p (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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{5} + 8 q^{7} - 64 q^{14} - 32 q^{16} + 8 q^{17} + 60 q^{20} - 8 q^{23} - 16 q^{25} - 8 q^{28} - 32 q^{29} + 16 q^{31} + 40 q^{32} + 40 q^{35} - 16 q^{37} - 48 q^{38} - 8 q^{40} + 8 q^{43} - 136 q^{44} + 40 q^{47} + 88 q^{50} + 8 q^{52} - 24 q^{53} + 24 q^{55} + 40 q^{58} - 8 q^{59} - 24 q^{61} + 64 q^{62} + 4 q^{65} - 16 q^{67} - 32 q^{68} + 48 q^{70} - 32 q^{73} - 80 q^{74} + 64 q^{76} - 24 q^{77} + 108 q^{80} - 16 q^{82} - 32 q^{83} - 8 q^{91} + 48 q^{92} + 72 q^{97} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.94606 − 1.94606i 0 5.57427i −0.730278 − 2.11346i 0 0.381738 − 0.381738i 6.95573 − 6.95573i 0 −2.69174 + 5.53406i
53.2 −1.63900 − 1.63900i 0 3.37267i −1.33214 − 1.79594i 0 2.74296 − 2.74296i 2.24982 − 2.24982i 0 −0.760162 + 5.12694i
53.3 −1.43528 − 1.43528i 0 2.12003i 1.71881 + 1.43027i 0 3.53238 − 3.53238i 0.172281 − 0.172281i 0 −0.414139 − 4.51981i
53.4 −0.906984 − 0.906984i 0 − 0.354760i 1.81576 + 1.30499i 0 −0.605706 + 0.605706i −2.13573 + 2.13573i 0 −0.463258 − 2.83048i
53.5 −0.611620 − 0.611620i 0 − 1.25184i −0.355410 − 2.20764i 0 −1.53695 + 1.53695i −1.98889 + 1.98889i 0 −1.13286 + 1.56761i
53.6 −0.398315 − 0.398315i 0 − 1.68269i −1.05498 + 1.97155i 0 0.577121 − 0.577121i −1.46687 + 1.46687i 0 1.20551 − 0.365087i
53.7 0.147011 + 0.147011i 0 − 1.95678i −0.203213 + 2.22681i 0 3.08655 − 3.08655i 0.581690 − 0.581690i 0 −0.357241 + 0.297492i
53.8 0.616723 + 0.616723i 0 − 1.23930i 2.22281 + 0.243174i 0 −0.0656273 + 0.0656273i 1.99775 − 1.99775i 0 1.22088 + 1.52083i
53.9 1.23209 + 1.23209i 0 1.03608i 2.23585 − 0.0313061i 0 −0.124208 + 0.124208i 1.18763 − 1.18763i 0 2.79333 + 2.71619i
53.10 1.46741 + 1.46741i 0 2.30658i 0.871670 − 2.05917i 0 0.885503 − 0.885503i −0.449871 + 0.449871i 0 4.30074 − 1.74255i
53.11 1.69552 + 1.69552i 0 3.74954i −2.02804 + 0.941842i 0 −1.89690 + 1.89690i −2.96638 + 2.96638i 0 −5.03547 − 1.84166i
53.12 1.77851 + 1.77851i 0 4.32620i −1.16084 − 1.91114i 0 −2.97686 + 2.97686i −4.13716 + 4.13716i 0 1.33440 − 5.46355i
287.1 −1.94606 + 1.94606i 0 − 5.57427i −0.730278 + 2.11346i 0 0.381738 + 0.381738i 6.95573 + 6.95573i 0 −2.69174 − 5.53406i
287.2 −1.63900 + 1.63900i 0 − 3.37267i −1.33214 + 1.79594i 0 2.74296 + 2.74296i 2.24982 + 2.24982i 0 −0.760162 − 5.12694i
287.3 −1.43528 + 1.43528i 0 − 2.12003i 1.71881 − 1.43027i 0 3.53238 + 3.53238i 0.172281 + 0.172281i 0 −0.414139 + 4.51981i
287.4 −0.906984 + 0.906984i 0 0.354760i 1.81576 − 1.30499i 0 −0.605706 − 0.605706i −2.13573 − 2.13573i 0 −0.463258 + 2.83048i
287.5 −0.611620 + 0.611620i 0 1.25184i −0.355410 + 2.20764i 0 −1.53695 − 1.53695i −1.98889 − 1.98889i 0 −1.13286 − 1.56761i
287.6 −0.398315 + 0.398315i 0 1.68269i −1.05498 − 1.97155i 0 0.577121 + 0.577121i −1.46687 − 1.46687i 0 1.20551 + 0.365087i
287.7 0.147011 − 0.147011i 0 1.95678i −0.203213 − 2.22681i 0 3.08655 + 3.08655i 0.581690 + 0.581690i 0 −0.357241 − 0.297492i
287.8 0.616723 − 0.616723i 0 1.23930i 2.22281 − 0.243174i 0 −0.0656273 − 0.0656273i 1.99775 + 1.99775i 0 1.22088 − 1.52083i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.p.b yes 24
3.b odd 2 1 585.2.p.a ✓ 24
5.c odd 4 1 585.2.p.a ✓ 24
15.e even 4 1 inner 585.2.p.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.p.a ✓ 24 3.b odd 2 1
585.2.p.a ✓ 24 5.c odd 4 1
585.2.p.b yes 24 1.a even 1 1 trivial
585.2.p.b yes 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 104 T_{2}^{20} - 8 T_{2}^{19} + 80 T_{2}^{17} + 3522 T_{2}^{16} - 192 T_{2}^{15} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display