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Properties

Label 225.3.l.a
Level $225$
Weight $3$
Character orbit 225.l
Analytic conductor $6.131$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,3,Mod(44,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.44");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.l (of order \(10\), 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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{4} + 28 q^{10} - 200 q^{16} + 60 q^{19} + 80 q^{22} + 184 q^{25} - 240 q^{28} + 120 q^{31} - 220 q^{34} - 60 q^{37} - 188 q^{40} + 440 q^{46} - 360 q^{49} + 700 q^{52} + 72 q^{55} + 120 q^{58} - 240 q^{61} - 880 q^{64} - 280 q^{67} - 984 q^{70} + 280 q^{73} - 200 q^{76} + 200 q^{79} + 1232 q^{85} + 3400 q^{88} - 500 q^{91} + 420 q^{94} + 280 q^{97}+O(q^{100}) \) 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 −1.17659 − 3.62117i 0 −8.49241 + 6.17010i −4.96650 − 0.577843i 0 5.51469i 20.0136 + 14.5408i 0 3.75106 + 18.6644i
44.2 −1.06460 − 3.27651i 0 −6.36607 + 4.62522i 4.57553 + 2.01606i 0 − 10.0440i 10.7833 + 7.83451i 0 1.73452 − 17.1381i
44.3 −0.967720 − 2.97834i 0 −4.69793 + 3.41325i −1.32066 − 4.82243i 0 5.58944i 4.57798 + 3.32610i 0 −13.0848 + 8.60014i
44.4 −0.811324 − 2.49700i 0 −2.34069 + 1.70061i −0.560638 + 4.96847i 0 − 9.88164i −2.35082 − 1.70797i 0 12.8611 − 2.63113i
44.5 −0.799623 − 2.46099i 0 −2.18099 + 1.58458i 4.81967 + 1.33071i 0 9.37453i −2.73016 − 1.98358i 0 −0.579068 − 12.9252i
44.6 −0.781339 − 2.40471i 0 −1.93609 + 1.40665i −2.40957 + 4.38109i 0 5.17510i −3.28694 − 2.38810i 0 12.4180 + 2.37121i
44.7 −0.474962 − 1.46178i 0 1.32485 − 0.962559i −1.74106 − 4.68708i 0 − 0.711296i −7.01017 − 5.09319i 0 −6.02455 + 4.77124i
44.8 −0.346103 − 1.06520i 0 2.22121 − 1.61381i 4.71895 − 1.65274i 0 − 7.00253i −6.11223 − 4.44079i 0 −3.39373 − 4.45458i
44.9 −0.142147 − 0.437482i 0 3.06488 − 2.22677i −4.99721 + 0.167025i 0 − 5.78044i −2.89841 − 2.10582i 0 0.783407 + 2.16245i
44.10 −0.0394141 − 0.121304i 0 3.22291 − 2.34158i −2.48570 + 4.33835i 0 12.4684i −0.823822 − 0.598542i 0 0.624232 + 0.130534i
44.11 0.0394141 + 0.121304i 0 3.22291 − 2.34158i 2.48570 − 4.33835i 0 12.4684i 0.823822 + 0.598542i 0 0.624232 + 0.130534i
44.12 0.142147 + 0.437482i 0 3.06488 − 2.22677i 4.99721 − 0.167025i 0 − 5.78044i 2.89841 + 2.10582i 0 0.783407 + 2.16245i
44.13 0.346103 + 1.06520i 0 2.22121 − 1.61381i −4.71895 + 1.65274i 0 − 7.00253i 6.11223 + 4.44079i 0 −3.39373 − 4.45458i
44.14 0.474962 + 1.46178i 0 1.32485 − 0.962559i 1.74106 + 4.68708i 0 − 0.711296i 7.01017 + 5.09319i 0 −6.02455 + 4.77124i
44.15 0.781339 + 2.40471i 0 −1.93609 + 1.40665i 2.40957 − 4.38109i 0 5.17510i 3.28694 + 2.38810i 0 12.4180 + 2.37121i
44.16 0.799623 + 2.46099i 0 −2.18099 + 1.58458i −4.81967 − 1.33071i 0 9.37453i 2.73016 + 1.98358i 0 −0.579068 − 12.9252i
44.17 0.811324 + 2.49700i 0 −2.34069 + 1.70061i 0.560638 − 4.96847i 0 − 9.88164i 2.35082 + 1.70797i 0 12.8611 − 2.63113i
44.18 0.967720 + 2.97834i 0 −4.69793 + 3.41325i 1.32066 + 4.82243i 0 5.58944i −4.57798 − 3.32610i 0 −13.0848 + 8.60014i
44.19 1.06460 + 3.27651i 0 −6.36607 + 4.62522i −4.57553 − 2.01606i 0 − 10.0440i −10.7833 − 7.83451i 0 1.73452 − 17.1381i
44.20 1.17659 + 3.62117i 0 −8.49241 + 6.17010i 4.96650 + 0.577843i 0 5.51469i −20.0136 − 14.5408i 0 3.75106 + 18.6644i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.l.a ✓ 80
3.b odd 2 1 inner 225.3.l.a ✓ 80
25.e even 10 1 inner 225.3.l.a ✓ 80
75.h odd 10 1 inner 225.3.l.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.l.a ✓ 80 1.a even 1 1 trivial
225.3.l.a ✓ 80 3.b odd 2 1 inner
225.3.l.a ✓ 80 25.e even 10 1 inner
225.3.l.a ✓ 80 75.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(225, [\chi])\).