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Properties

Label 891.2.k.a
Level $891$
Weight $2$
Character orbit 891.k
Analytic conductor $7.115$
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] = [891,2,Mod(161,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.k (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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 99)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 10 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 10 q^{4} + 10 q^{7} + 10 q^{13} - 10 q^{16} - 50 q^{19} + 22 q^{22} + 4 q^{25} - 20 q^{28} + 12 q^{31} + 20 q^{34} - 6 q^{37} - 30 q^{40} - 40 q^{46} + 2 q^{49} + 10 q^{52} - 18 q^{55} + 58 q^{58} + 10 q^{61} - 8 q^{64} - 20 q^{67} - 60 q^{70} - 20 q^{73} + 10 q^{79} - 2 q^{82} + 10 q^{85} + 118 q^{88} + 52 q^{91} + 10 q^{94} - 54 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 −2.17940 − 1.58343i 0 1.62452 + 4.99975i −0.850459 − 1.17056i 0 1.37298 − 0.446107i 2.71136 − 8.34470i 0 3.89775i
161.2 −1.89658 − 1.37794i 0 1.08024 + 3.32465i −1.71732 − 2.36369i 0 −1.25106 + 0.406493i 1.08356 − 3.33484i 0 6.84930i
161.3 −1.71719 − 1.24761i 0 0.774180 + 2.38268i 1.75074 + 2.40969i 0 3.03899 − 0.987427i 0.331430 − 1.02004i 0 − 6.32217i
161.4 −1.59313 − 1.15748i 0 0.580279 + 1.78592i 1.16215 + 1.59956i 0 −0.301636 + 0.0980075i −0.0743474 + 0.228818i 0 − 3.89347i
161.5 −1.46797 − 1.06654i 0 0.399387 + 1.22919i 0.706314 + 0.972158i 0 −4.60425 + 1.49601i −0.396737 + 1.22103i 0 − 2.18041i
161.6 −1.16363 − 0.845427i 0 0.0212543 + 0.0654141i −2.09686 − 2.88608i 0 2.05824 − 0.668764i −0.858363 + 2.64177i 0 5.13106i
161.7 −0.931921 − 0.677080i 0 −0.207995 − 0.640143i −0.551104 − 0.758530i 0 −0.824408 + 0.267866i −0.951517 + 2.92847i 0 1.08003i
161.8 −0.505702 − 0.367414i 0 −0.497292 − 1.53051i 0.749839 + 1.03206i 0 4.60560 − 1.49645i −0.697171 + 2.14567i 0 − 0.797419i
161.9 −0.417544 − 0.303364i 0 −0.535720 − 1.64878i 0.956312 + 1.31625i 0 −0.785572 + 0.255248i −0.595467 + 1.83266i 0 − 0.839703i
161.10 −0.299321 − 0.217469i 0 −0.575734 − 1.77193i −1.39685 − 1.92260i 0 −1.49987 + 0.487337i −0.441671 + 1.35932i 0 0.879247i
161.11 0.299321 + 0.217469i 0 −0.575734 − 1.77193i 1.39685 + 1.92260i 0 −1.49987 + 0.487337i 0.441671 − 1.35932i 0 0.879247i
161.12 0.417544 + 0.303364i 0 −0.535720 − 1.64878i −0.956312 − 1.31625i 0 −0.785572 + 0.255248i 0.595467 − 1.83266i 0 − 0.839703i
161.13 0.505702 + 0.367414i 0 −0.497292 − 1.53051i −0.749839 − 1.03206i 0 4.60560 − 1.49645i 0.697171 − 2.14567i 0 − 0.797419i
161.14 0.931921 + 0.677080i 0 −0.207995 − 0.640143i 0.551104 + 0.758530i 0 −0.824408 + 0.267866i 0.951517 − 2.92847i 0 1.08003i
161.15 1.16363 + 0.845427i 0 0.0212543 + 0.0654141i 2.09686 + 2.88608i 0 2.05824 − 0.668764i 0.858363 − 2.64177i 0 5.13106i
161.16 1.46797 + 1.06654i 0 0.399387 + 1.22919i −0.706314 − 0.972158i 0 −4.60425 + 1.49601i 0.396737 − 1.22103i 0 − 2.18041i
161.17 1.59313 + 1.15748i 0 0.580279 + 1.78592i −1.16215 − 1.59956i 0 −0.301636 + 0.0980075i 0.0743474 − 0.228818i 0 − 3.89347i
161.18 1.71719 + 1.24761i 0 0.774180 + 2.38268i −1.75074 − 2.40969i 0 3.03899 − 0.987427i −0.331430 + 1.02004i 0 − 6.32217i
161.19 1.89658 + 1.37794i 0 1.08024 + 3.32465i 1.71732 + 2.36369i 0 −1.25106 + 0.406493i −1.08356 + 3.33484i 0 6.84930i
161.20 2.17940 + 1.58343i 0 1.62452 + 4.99975i 0.850459 + 1.17056i 0 1.37298 − 0.446107i −2.71136 + 8.34470i 0 3.89775i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.k.a 80
3.b odd 2 1 inner 891.2.k.a 80
9.c even 3 1 99.2.p.a ✓ 80
9.c even 3 1 297.2.t.a 80
9.d odd 6 1 99.2.p.a ✓ 80
9.d odd 6 1 297.2.t.a 80
11.d odd 10 1 inner 891.2.k.a 80
33.f even 10 1 inner 891.2.k.a 80
99.o odd 30 1 99.2.p.a ✓ 80
99.o odd 30 1 297.2.t.a 80
99.p even 30 1 99.2.p.a ✓ 80
99.p even 30 1 297.2.t.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.p.a ✓ 80 9.c even 3 1
99.2.p.a ✓ 80 9.d odd 6 1
99.2.p.a ✓ 80 99.o odd 30 1
99.2.p.a ✓ 80 99.p even 30 1
297.2.t.a 80 9.c even 3 1
297.2.t.a 80 9.d odd 6 1
297.2.t.a 80 99.o odd 30 1
297.2.t.a 80 99.p even 30 1
891.2.k.a 80 1.a even 1 1 trivial
891.2.k.a 80 3.b odd 2 1 inner
891.2.k.a 80 11.d odd 10 1 inner
891.2.k.a 80 33.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 25 T_{2}^{78} + 385 T_{2}^{76} + 4688 T_{2}^{74} + 49104 T_{2}^{72} + 439655 T_{2}^{70} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display