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Properties

Label 441.2.bb.b
Level $441$
Weight $2$
Character orbit 441.bb
Analytic conductor $3.521$
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] = [441,2,Mod(37,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bb (of order \(21\), degree \(12\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{4} + 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{4} + 28 q^{7} - 42 q^{13} + 36 q^{16} + 28 q^{19} - 14 q^{22} + 32 q^{25} - 14 q^{28} + 56 q^{31} - 28 q^{34} - 2 q^{37} - 42 q^{40} - 6 q^{43} - 56 q^{46} + 28 q^{49} + 42 q^{52} + 98 q^{55} - 14 q^{58} - 14 q^{61} + 46 q^{64} + 36 q^{67} - 70 q^{70} + 70 q^{73} - 98 q^{76} - 54 q^{79} - 140 q^{82} + 28 q^{85} + 28 q^{88} - 70 q^{91} - 70 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.56648 + 1.06801i 0 0.582537 − 1.48428i −1.18685 − 0.366095i 0 2.57942 + 0.588735i −0.171072 − 0.749515i 0 2.25017 − 0.694086i
37.2 1.56648 − 1.06801i 0 0.582537 − 1.48428i 1.18685 + 0.366095i 0 2.57942 + 0.588735i 0.171072 + 0.749515i 0 2.25017 − 0.694086i
46.1 −0.613977 + 1.56439i 0 −0.604237 − 0.560650i 3.47262 + 2.36759i 0 −1.14795 + 2.38374i −1.78020 + 0.857299i 0 −5.83593 + 3.97887i
46.2 0.613977 − 1.56439i 0 −0.604237 − 0.560650i −3.47262 − 2.36759i 0 −1.14795 + 2.38374i 1.78020 − 0.857299i 0 −5.83593 + 3.97887i
100.1 −0.511849 − 0.157884i 0 −1.41542 − 0.965014i −4.32038 + 0.651193i 0 2.06853 + 1.64960i 1.24006 + 1.55498i 0 2.31420 + 0.348809i
100.2 0.511849 + 0.157884i 0 −1.41542 − 0.965014i 4.32038 − 0.651193i 0 2.06853 + 1.64960i −1.24006 − 1.55498i 0 2.31420 + 0.348809i
109.1 −1.40321 + 1.30199i 0 0.124363 − 1.65951i 0.913329 − 2.32712i 0 2.06853 + 1.64960i −0.400813 − 0.502603i 0 1.74829 + 4.45458i
109.2 1.40321 − 1.30199i 0 0.124363 − 1.65951i −0.913329 + 2.32712i 0 2.06853 + 1.64960i 0.400813 + 0.502603i 0 1.74829 + 4.45458i
163.1 −0.613977 − 1.56439i 0 −0.604237 + 0.560650i 3.47262 − 2.36759i 0 −1.14795 − 2.38374i −1.78020 − 0.857299i 0 −5.83593 − 3.97887i
163.2 0.613977 + 1.56439i 0 −0.604237 + 0.560650i −3.47262 + 2.36759i 0 −1.14795 − 2.38374i 1.78020 + 0.857299i 0 −5.83593 − 3.97887i
172.1 −0.511849 + 0.157884i 0 −1.41542 + 0.965014i −4.32038 − 0.651193i 0 2.06853 − 1.64960i 1.24006 − 1.55498i 0 2.31420 − 0.348809i
172.2 0.511849 − 0.157884i 0 −1.41542 + 0.965014i 4.32038 + 0.651193i 0 2.06853 − 1.64960i −1.24006 + 1.55498i 0 2.31420 − 0.348809i
235.1 −2.32588 − 0.350570i 0 3.37568 + 1.04126i 0.248330 + 3.31374i 0 −1.14795 − 2.38374i −3.24795 − 1.56413i 0 0.584111 − 7.79442i
235.2 2.32588 + 0.350570i 0 3.37568 + 1.04126i −0.248330 − 3.31374i 0 −1.14795 − 2.38374i 3.24795 + 1.56413i 0 0.584111 − 7.79442i
289.1 −2.32588 + 0.350570i 0 3.37568 − 1.04126i 0.248330 − 3.31374i 0 −1.14795 + 2.38374i −3.24795 + 1.56413i 0 0.584111 + 7.79442i
289.2 2.32588 − 0.350570i 0 3.37568 − 1.04126i −0.248330 + 3.31374i 0 −1.14795 + 2.38374i 3.24795 − 1.56413i 0 0.584111 + 7.79442i
298.1 −1.56648 − 1.06801i 0 0.582537 + 1.48428i −1.18685 + 0.366095i 0 2.57942 − 0.588735i −0.171072 + 0.749515i 0 2.25017 + 0.694086i
298.2 1.56648 + 1.06801i 0 0.582537 + 1.48428i 1.18685 − 0.366095i 0 2.57942 − 0.588735i 0.171072 − 0.749515i 0 2.25017 + 0.694086i
352.1 −1.40321 − 1.30199i 0 0.124363 + 1.65951i 0.913329 + 2.32712i 0 2.06853 − 1.64960i −0.400813 + 0.502603i 0 1.74829 − 4.45458i
352.2 1.40321 + 1.30199i 0 0.124363 + 1.65951i −0.913329 − 2.32712i 0 2.06853 − 1.64960i 0.400813 − 0.502603i 0 1.74829 − 4.45458i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.g even 21 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bb.b ✓ 24
3.b odd 2 1 inner 441.2.bb.b ✓ 24
49.g even 21 1 inner 441.2.bb.b ✓ 24
147.n odd 42 1 inner 441.2.bb.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.bb.b ✓ 24 1.a even 1 1 trivial
441.2.bb.b ✓ 24 3.b odd 2 1 inner
441.2.bb.b ✓ 24 49.g even 21 1 inner
441.2.bb.b ✓ 24 147.n odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 28 T_{2}^{20} + 35 T_{2}^{18} - 119 T_{2}^{16} - 511 T_{2}^{14} + 10059 T_{2}^{12} + \cdots + 90601 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display