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Properties

Label 405.2.m.e
Level $405$
Weight $2$
Character orbit 405.m
Analytic conductor $3.234$
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] = [405,2,Mod(53,405)]
 
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([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.m (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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7} + 12 q^{13} + 12 q^{16} - 60 q^{22} - 24 q^{25} + 60 q^{34} + 36 q^{37} + 84 q^{40} - 36 q^{43} - 48 q^{46} - 72 q^{49} + 84 q^{52} + 84 q^{58} + 24 q^{67} - 120 q^{70} + 12 q^{73} - 24 q^{76} + 48 q^{79} - 96 q^{82} - 48 q^{85} - 168 q^{88} - 48 q^{91} - 48 q^{94} - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −0.579996 + 2.16457i 0 −2.61693 − 1.51089i −0.820748 + 2.07999i 0 3.45159 + 0.924851i 1.61908 − 1.61908i 0 −4.02627 − 2.98296i
53.2 −0.488394 + 1.82271i 0 −1.35170 − 0.780404i −1.40788 − 1.73720i 0 −0.853795 − 0.228774i −0.586021 + 0.586021i 0 3.85402 − 1.71773i
53.3 −0.167218 + 0.624065i 0 1.37056 + 0.791291i 1.08590 − 1.95469i 0 −0.231769 − 0.0621024i −1.63669 + 1.63669i 0 1.03827 + 1.00453i
53.4 0.167218 − 0.624065i 0 1.37056 + 0.791291i −1.08590 + 1.95469i 0 −0.231769 − 0.0621024i 1.63669 − 1.63669i 0 1.03827 + 1.00453i
53.5 0.488394 − 1.82271i 0 −1.35170 − 0.780404i 1.40788 + 1.73720i 0 −0.853795 − 0.228774i 0.586021 − 0.586021i 0 3.85402 − 1.71773i
53.6 0.579996 − 2.16457i 0 −2.61693 − 1.51089i 0.820748 − 2.07999i 0 3.45159 + 0.924851i −1.61908 + 1.61908i 0 −4.02627 − 2.98296i
107.1 −0.579996 − 2.16457i 0 −2.61693 + 1.51089i −0.820748 − 2.07999i 0 3.45159 − 0.924851i 1.61908 + 1.61908i 0 −4.02627 + 2.98296i
107.2 −0.488394 − 1.82271i 0 −1.35170 + 0.780404i −1.40788 + 1.73720i 0 −0.853795 + 0.228774i −0.586021 − 0.586021i 0 3.85402 + 1.71773i
107.3 −0.167218 − 0.624065i 0 1.37056 − 0.791291i 1.08590 + 1.95469i 0 −0.231769 + 0.0621024i −1.63669 − 1.63669i 0 1.03827 − 1.00453i
107.4 0.167218 + 0.624065i 0 1.37056 − 0.791291i −1.08590 − 1.95469i 0 −0.231769 + 0.0621024i 1.63669 + 1.63669i 0 1.03827 − 1.00453i
107.5 0.488394 + 1.82271i 0 −1.35170 + 0.780404i 1.40788 − 1.73720i 0 −0.853795 + 0.228774i 0.586021 + 0.586021i 0 3.85402 + 1.71773i
107.6 0.579996 + 2.16457i 0 −2.61693 + 1.51089i 0.820748 + 2.07999i 0 3.45159 − 0.924851i −1.61908 − 1.61908i 0 −4.02627 + 2.98296i
188.1 −2.53849 + 0.680187i 0 4.24923 − 2.45330i −0.0837902 − 2.23450i 0 0.177348 + 0.661870i −5.40134 + 5.40134i 0 1.73258 + 5.61526i
188.2 −1.38490 + 0.371084i 0 0.0482034 − 0.0278302i 2.19367 − 0.433384i 0 1.26950 + 4.73784i 1.97121 − 1.97121i 0 −2.87720 + 1.41423i
188.3 −0.187662 + 0.0502840i 0 −1.69936 + 0.981127i −1.82917 + 1.28613i 0 −0.812874 − 3.03369i 0.544328 − 0.544328i 0 0.278595 − 0.333337i
188.4 0.187662 − 0.0502840i 0 −1.69936 + 0.981127i 1.82917 − 1.28613i 0 −0.812874 − 3.03369i −0.544328 + 0.544328i 0 0.278595 − 0.333337i
188.5 1.38490 − 0.371084i 0 0.0482034 − 0.0278302i −2.19367 + 0.433384i 0 1.26950 + 4.73784i −1.97121 + 1.97121i 0 −2.87720 + 1.41423i
188.6 2.53849 − 0.680187i 0 4.24923 − 2.45330i 0.0837902 + 2.23450i 0 0.177348 + 0.661870i 5.40134 − 5.40134i 0 1.73258 + 5.61526i
377.1 −2.53849 − 0.680187i 0 4.24923 + 2.45330i −0.0837902 + 2.23450i 0 0.177348 − 0.661870i −5.40134 − 5.40134i 0 1.73258 − 5.61526i
377.2 −1.38490 − 0.371084i 0 0.0482034 + 0.0278302i 2.19367 + 0.433384i 0 1.26950 − 4.73784i 1.97121 + 1.97121i 0 −2.87720 − 1.41423i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 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 405.2.m.e 24
3.b odd 2 1 inner 405.2.m.e 24
5.c odd 4 1 405.2.m.d 24
9.c even 3 1 405.2.f.b ✓ 24
9.c even 3 1 405.2.m.d 24
9.d odd 6 1 405.2.f.b ✓ 24
9.d odd 6 1 405.2.m.d 24
15.e even 4 1 405.2.m.d 24
45.k odd 12 1 405.2.f.b ✓ 24
45.k odd 12 1 inner 405.2.m.e 24
45.l even 12 1 405.2.f.b ✓ 24
45.l even 12 1 inner 405.2.m.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.f.b ✓ 24 9.c even 3 1
405.2.f.b ✓ 24 9.d odd 6 1
405.2.f.b ✓ 24 45.k odd 12 1
405.2.f.b ✓ 24 45.l even 12 1
405.2.m.d 24 5.c odd 4 1
405.2.m.d 24 9.c even 3 1
405.2.m.d 24 9.d odd 6 1
405.2.m.d 24 15.e even 4 1
405.2.m.e 24 1.a even 1 1 trivial
405.2.m.e 24 3.b odd 2 1 inner
405.2.m.e 24 45.k odd 12 1 inner
405.2.m.e 24 45.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 45 T_{2}^{20} + 1785 T_{2}^{16} + 4152 T_{2}^{14} - 7720 T_{2}^{12} - 23040 T_{2}^{10} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display