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Properties

Label 585.2.q.a
Level $585$
Weight $2$
Character orbit 585.q
Analytic conductor $4.671$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(44,585)]
 
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, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.44");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.q (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: \(56\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 88 q^{16} + 32 q^{19} - 24 q^{34} + 32 q^{40} + 96 q^{46} - 32 q^{55} + 64 q^{61} + 56 q^{70} - 128 q^{76} + 48 q^{79} - 56 q^{85} - 80 q^{91} + 64 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −1.96833 + 1.96833i 0 − 5.74861i −0.820542 − 2.08007i 0 1.68108 − 1.68108i 7.37849 + 7.37849i 0 5.70936 + 2.47917i
44.2 −1.96833 + 1.96833i 0 − 5.74861i 2.08007 + 0.820542i 0 −1.68108 + 1.68108i 7.37849 + 7.37849i 0 −5.70936 + 2.47917i
44.3 −1.66271 + 1.66271i 0 − 3.52920i −2.06619 + 0.854908i 0 1.64823 − 1.64823i 2.54262 + 2.54262i 0 2.01401 − 4.85693i
44.4 −1.66271 + 1.66271i 0 − 3.52920i −0.854908 + 2.06619i 0 −1.64823 + 1.64823i 2.54262 + 2.54262i 0 −2.01401 − 4.85693i
44.5 −1.18072 + 1.18072i 0 − 0.788211i 1.05170 − 1.97330i 0 −1.31579 + 1.31579i −1.43079 − 1.43079i 0 1.08816 + 3.57169i
44.6 −1.18072 + 1.18072i 0 − 0.788211i 1.97330 − 1.05170i 0 1.31579 − 1.31579i −1.43079 − 1.43079i 0 −1.08816 + 3.57169i
44.7 −1.14885 + 1.14885i 0 − 0.639696i −1.66746 − 1.48983i 0 −1.19865 + 1.19865i −1.56278 − 1.56278i 0 3.62724 − 0.204067i
44.8 −1.14885 + 1.14885i 0 − 0.639696i 1.48983 + 1.66746i 0 1.19865 − 1.19865i −1.56278 − 1.56278i 0 −3.62724 − 0.204067i
44.9 −0.968906 + 0.968906i 0 0.122441i −1.80686 + 1.31730i 0 −3.34174 + 3.34174i −2.05645 − 2.05645i 0 0.474338 − 3.02701i
44.10 −0.968906 + 0.968906i 0 0.122441i −1.31730 + 1.80686i 0 3.34174 − 3.34174i −2.05645 − 2.05645i 0 −0.474338 − 3.02701i
44.11 −0.410110 + 0.410110i 0 1.66362i −1.96644 − 1.06449i 0 1.89766 − 1.89766i −1.50249 − 1.50249i 0 1.24301 − 0.369899i
44.12 −0.410110 + 0.410110i 0 1.66362i 1.06449 + 1.96644i 0 −1.89766 + 1.89766i −1.50249 − 1.50249i 0 −1.24301 − 0.369899i
44.13 −0.200424 + 0.200424i 0 1.91966i −0.196474 − 2.22742i 0 −1.87641 + 1.87641i −0.785594 − 0.785594i 0 0.485806 + 0.407050i
44.14 −0.200424 + 0.200424i 0 1.91966i 2.22742 + 0.196474i 0 1.87641 − 1.87641i −0.785594 − 0.785594i 0 −0.485806 + 0.407050i
44.15 0.200424 − 0.200424i 0 1.91966i −2.22742 − 0.196474i 0 1.87641 − 1.87641i 0.785594 + 0.785594i 0 −0.485806 + 0.407050i
44.16 0.200424 − 0.200424i 0 1.91966i 0.196474 + 2.22742i 0 −1.87641 + 1.87641i 0.785594 + 0.785594i 0 0.485806 + 0.407050i
44.17 0.410110 − 0.410110i 0 1.66362i −1.06449 − 1.96644i 0 −1.89766 + 1.89766i 1.50249 + 1.50249i 0 −1.24301 − 0.369899i
44.18 0.410110 − 0.410110i 0 1.66362i 1.96644 + 1.06449i 0 1.89766 − 1.89766i 1.50249 + 1.50249i 0 1.24301 − 0.369899i
44.19 0.968906 − 0.968906i 0 0.122441i 1.31730 − 1.80686i 0 3.34174 − 3.34174i 2.05645 + 2.05645i 0 −0.474338 − 3.02701i
44.20 0.968906 − 0.968906i 0 0.122441i 1.80686 − 1.31730i 0 −3.34174 + 3.34174i 2.05645 + 2.05645i 0 0.474338 − 3.02701i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.d odd 4 1 inner
15.d odd 2 1 inner
39.f even 4 1 inner
65.g odd 4 1 inner
195.n 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.q.a ✓ 56
3.b odd 2 1 inner 585.2.q.a ✓ 56
5.b even 2 1 inner 585.2.q.a ✓ 56
13.d odd 4 1 inner 585.2.q.a ✓ 56
15.d odd 2 1 inner 585.2.q.a ✓ 56
39.f even 4 1 inner 585.2.q.a ✓ 56
65.g odd 4 1 inner 585.2.q.a ✓ 56
195.n even 4 1 inner 585.2.q.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.q.a ✓ 56 1.a even 1 1 trivial
585.2.q.a ✓ 56 3.b odd 2 1 inner
585.2.q.a ✓ 56 5.b even 2 1 inner
585.2.q.a ✓ 56 13.d odd 4 1 inner
585.2.q.a ✓ 56 15.d odd 2 1 inner
585.2.q.a ✓ 56 39.f even 4 1 inner
585.2.q.a ✓ 56 65.g odd 4 1 inner
585.2.q.a ✓ 56 195.n even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).