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Properties

Label 405.3.l.n
Level $405$
Weight $3$
Character orbit 405.l
Analytic conductor $11.035$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(28,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{7} + 80 q^{10} - 40 q^{13} + 152 q^{16} + 136 q^{22} + 32 q^{25} - 224 q^{28} - 200 q^{31} + 32 q^{37} + 48 q^{40} - 136 q^{43} + 304 q^{46} - 640 q^{52} + 496 q^{55} - 48 q^{58} + 280 q^{61}+ \cdots - 448 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −3.68016 − 0.986097i 0 9.10712 + 5.25800i −4.36446 + 2.43957i 0 1.55546 + 0.416783i −17.5546 − 17.5546i 0 18.4676 − 4.67423i
28.2 −2.78913 − 0.747344i 0 3.75659 + 2.16887i 4.50635 − 2.16628i 0 −11.5096 − 3.08399i −0.689596 − 0.689596i 0 −14.1877 + 2.67423i
28.3 −1.43685 − 0.385003i 0 −1.54779 − 0.893616i −4.97194 − 0.528949i 0 7.10453 + 1.90365i 6.08729 + 6.08729i 0 6.94029 + 2.67423i
28.4 −0.933250 − 0.250064i 0 −2.65568 − 1.53326i 2.47196 + 4.34619i 0 −2.61447 − 0.700546i 4.82775 + 4.82775i 0 −1.22013 − 4.67423i
28.5 0.933250 + 0.250064i 0 −2.65568 − 1.53326i −2.47196 − 4.34619i 0 −2.61447 − 0.700546i −4.82775 − 4.82775i 0 −1.22013 − 4.67423i
28.6 1.43685 + 0.385003i 0 −1.54779 − 0.893616i 4.97194 + 0.528949i 0 7.10453 + 1.90365i −6.08729 − 6.08729i 0 6.94029 + 2.67423i
28.7 2.78913 + 0.747344i 0 3.75659 + 2.16887i −4.50635 + 2.16628i 0 −11.5096 − 3.08399i 0.689596 + 0.689596i 0 −14.1877 + 2.67423i
28.8 3.68016 + 0.986097i 0 9.10712 + 5.25800i 4.36446 − 2.43957i 0 1.55546 + 0.416783i 17.5546 + 17.5546i 0 18.4676 − 4.67423i
217.1 −3.68016 + 0.986097i 0 9.10712 − 5.25800i −4.36446 − 2.43957i 0 1.55546 − 0.416783i −17.5546 + 17.5546i 0 18.4676 + 4.67423i
217.2 −2.78913 + 0.747344i 0 3.75659 − 2.16887i 4.50635 + 2.16628i 0 −11.5096 + 3.08399i −0.689596 + 0.689596i 0 −14.1877 − 2.67423i
217.3 −1.43685 + 0.385003i 0 −1.54779 + 0.893616i −4.97194 + 0.528949i 0 7.10453 − 1.90365i 6.08729 − 6.08729i 0 6.94029 − 2.67423i
217.4 −0.933250 + 0.250064i 0 −2.65568 + 1.53326i 2.47196 − 4.34619i 0 −2.61447 + 0.700546i 4.82775 − 4.82775i 0 −1.22013 + 4.67423i
217.5 0.933250 − 0.250064i 0 −2.65568 + 1.53326i −2.47196 + 4.34619i 0 −2.61447 + 0.700546i −4.82775 + 4.82775i 0 −1.22013 + 4.67423i
217.6 1.43685 − 0.385003i 0 −1.54779 + 0.893616i 4.97194 − 0.528949i 0 7.10453 − 1.90365i −6.08729 + 6.08729i 0 6.94029 − 2.67423i
217.7 2.78913 − 0.747344i 0 3.75659 − 2.16887i −4.50635 − 2.16628i 0 −11.5096 + 3.08399i 0.689596 − 0.689596i 0 −14.1877 − 2.67423i
217.8 3.68016 − 0.986097i 0 9.10712 − 5.25800i 4.36446 + 2.43957i 0 1.55546 − 0.416783i 17.5546 − 17.5546i 0 18.4676 + 4.67423i
298.1 −0.986097 − 3.68016i 0 −9.10712 + 5.25800i −0.0695010 + 4.99952i 0 −0.416783 − 1.55546i 17.5546 + 17.5546i 0 18.4676 − 4.67423i
298.2 −0.747344 − 2.78913i 0 −3.75659 + 2.16887i 0.377122 − 4.98576i 0 3.08399 + 11.5096i 0.689596 + 0.689596i 0 −14.1877 + 2.67423i
298.3 −0.385003 − 1.43685i 0 1.54779 − 0.893616i −2.94405 + 4.04135i 0 −1.90365 − 7.10453i −6.08729 − 6.08729i 0 6.94029 + 2.67423i
298.4 −0.250064 − 0.933250i 0 2.65568 − 1.53326i 4.99990 + 0.0323160i 0 0.700546 + 2.61447i −4.82775 − 4.82775i 0 −1.22013 − 4.67423i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.n 32
3.b odd 2 1 inner 405.3.l.n 32
5.c odd 4 1 inner 405.3.l.n 32
9.c even 3 1 135.3.g.b ✓ 16
9.c even 3 1 inner 405.3.l.n 32
9.d odd 6 1 135.3.g.b ✓ 16
9.d odd 6 1 inner 405.3.l.n 32
15.e even 4 1 inner 405.3.l.n 32
45.h odd 6 1 675.3.g.j 16
45.j even 6 1 675.3.g.j 16
45.k odd 12 1 135.3.g.b ✓ 16
45.k odd 12 1 inner 405.3.l.n 32
45.k odd 12 1 675.3.g.j 16
45.l even 12 1 135.3.g.b ✓ 16
45.l even 12 1 inner 405.3.l.n 32
45.l even 12 1 675.3.g.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.3.g.b ✓ 16 9.c even 3 1
135.3.g.b ✓ 16 9.d odd 6 1
135.3.g.b ✓ 16 45.k odd 12 1
135.3.g.b ✓ 16 45.l even 12 1
405.3.l.n 32 1.a even 1 1 trivial
405.3.l.n 32 3.b odd 2 1 inner
405.3.l.n 32 5.c odd 4 1 inner
405.3.l.n 32 9.c even 3 1 inner
405.3.l.n 32 9.d odd 6 1 inner
405.3.l.n 32 15.e even 4 1 inner
405.3.l.n 32 45.k odd 12 1 inner
405.3.l.n 32 45.l even 12 1 inner
675.3.g.j 16 45.h odd 6 1
675.3.g.j 16 45.j even 6 1
675.3.g.j 16 45.k odd 12 1
675.3.g.j 16 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 286 T_{2}^{28} + 65527 T_{2}^{24} - 4481566 T_{2}^{20} + 240112237 T_{2}^{16} + \cdots + 3906250000 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display