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Properties

Label 405.3.d.b
Level $405$
Weight $3$
Character orbit 405.d
Analytic conductor $11.035$
Analytic rank $0$
Dimension $24$
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] = [405,3,Mod(404,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.404");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.d (of order \(2\), degree \(1\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91} + 384 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
404.1 −3.57411 0 8.77429 3.06744 − 3.94852i 0 6.02185i −17.0638 0 −10.9634 + 14.1125i
404.2 −3.57411 0 8.77429 3.06744 + 3.94852i 0 − 6.02185i −17.0638 0 −10.9634 − 14.1125i
404.3 −3.48400 0 8.13825 −4.82085 − 1.32641i 0 − 12.7764i −14.4177 0 16.7958 + 4.62123i
404.4 −3.48400 0 8.13825 −4.82085 + 1.32641i 0 12.7764i −14.4177 0 16.7958 − 4.62123i
404.5 −2.28392 0 1.21631 −1.08806 − 4.88018i 0 3.44650i 6.35774 0 2.48505 + 11.1459i
404.6 −2.28392 0 1.21631 −1.08806 + 4.88018i 0 − 3.44650i 6.35774 0 2.48505 − 11.1459i
404.7 −2.12750 0 0.526255 4.15908 − 2.77526i 0 1.17536i 7.39039 0 −8.84844 + 5.90436i
404.8 −2.12750 0 0.526255 4.15908 + 2.77526i 0 − 1.17536i 7.39039 0 −8.84844 − 5.90436i
404.9 −1.15564 0 −2.66451 −3.41362 − 3.65339i 0 − 7.03988i 7.70174 0 3.94490 + 4.22198i
404.10 −1.15564 0 −2.66451 −3.41362 + 3.65339i 0 7.03988i 7.70174 0 3.94490 − 4.22198i
404.11 −0.0969975 0 −3.99059 4.26797 − 2.60469i 0 − 10.1824i 0.775068 0 −0.413983 + 0.252648i
404.12 −0.0969975 0 −3.99059 4.26797 + 2.60469i 0 10.1824i 0.775068 0 −0.413983 − 0.252648i
404.13 0.0969975 0 −3.99059 −4.26797 − 2.60469i 0 10.1824i −0.775068 0 −0.413983 − 0.252648i
404.14 0.0969975 0 −3.99059 −4.26797 + 2.60469i 0 − 10.1824i −0.775068 0 −0.413983 + 0.252648i
404.15 1.15564 0 −2.66451 3.41362 − 3.65339i 0 7.03988i −7.70174 0 3.94490 − 4.22198i
404.16 1.15564 0 −2.66451 3.41362 + 3.65339i 0 − 7.03988i −7.70174 0 3.94490 + 4.22198i
404.17 2.12750 0 0.526255 −4.15908 − 2.77526i 0 − 1.17536i −7.39039 0 −8.84844 − 5.90436i
404.18 2.12750 0 0.526255 −4.15908 + 2.77526i 0 1.17536i −7.39039 0 −8.84844 + 5.90436i
404.19 2.28392 0 1.21631 1.08806 − 4.88018i 0 − 3.44650i −6.35774 0 2.48505 − 11.1459i
404.20 2.28392 0 1.21631 1.08806 + 4.88018i 0 3.44650i −6.35774 0 2.48505 + 11.1459i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 404.24
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.d.b ✓ 24
3.b odd 2 1 inner 405.3.d.b ✓ 24
5.b even 2 1 inner 405.3.d.b ✓ 24
9.c even 3 2 405.3.h.k 48
9.d odd 6 2 405.3.h.k 48
15.d odd 2 1 inner 405.3.d.b ✓ 24
45.h odd 6 2 405.3.h.k 48
45.j even 6 2 405.3.h.k 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.3.d.b ✓ 24 1.a even 1 1 trivial
405.3.d.b ✓ 24 3.b odd 2 1 inner
405.3.d.b ✓ 24 5.b even 2 1 inner
405.3.d.b ✓ 24 15.d odd 2 1 inner
405.3.h.k 48 9.c even 3 2
405.3.h.k 48 9.d odd 6 2
405.3.h.k 48 45.h odd 6 2
405.3.h.k 48 45.j even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 36T_{2}^{10} + 468T_{2}^{8} - 2666T_{2}^{6} + 6489T_{2}^{4} - 4950T_{2}^{2} + 46 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display