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Properties

Label 891.2.n.l
Level $891$
Weight $2$
Character orbit 891.n
Analytic conductor $7.115$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(136,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.n (of order \(15\), degree \(8\), 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{4} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{4} + 14 q^{7} + 64 q^{10} + 14 q^{13} + 4 q^{16} - 16 q^{19} - 16 q^{22} + 20 q^{25} - 36 q^{28} + 4 q^{31} - 84 q^{34} - 24 q^{37} + 106 q^{40} - 84 q^{43} - 76 q^{46} + 54 q^{49} - 52 q^{52} - 72 q^{55} + 10 q^{58} + 6 q^{61} - 104 q^{64} - 76 q^{67} + 8 q^{70} - 92 q^{73} - 204 q^{76} + 102 q^{79} + 24 q^{82} - 54 q^{85} + 44 q^{91} + 118 q^{94} + 66 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −2.33604 + 1.04007i 0 3.03706 − 3.37300i 1.88024 + 0.837138i 0 −3.87187 − 0.822992i −2.00615 + 6.17430i 0 −5.26300
136.2 −2.26297 + 1.00754i 0 2.76762 − 3.07376i −2.42411 − 1.07928i 0 −0.234155 − 0.0497713i −1.63516 + 5.03251i 0 6.57309
136.3 −1.67544 + 0.745953i 0 0.912387 − 1.01331i −0.0824312 − 0.0367007i 0 4.60859 + 0.979587i 0.360704 − 1.11013i 0 0.165485
136.4 −1.24612 + 0.554810i 0 −0.0932499 + 0.103565i 1.89778 + 0.844946i 0 0.101465 + 0.0215671i 0.901773 − 2.77537i 0 −2.83366
136.5 −1.05062 + 0.467766i 0 −0.453266 + 0.503403i −3.57276 − 1.59069i 0 −4.48724 − 0.953792i 0.951502 − 2.92842i 0 4.49768
136.6 −0.575085 + 0.256045i 0 −1.07310 + 1.19179i −1.24862 − 0.555922i 0 0.575145 + 0.122251i 0.701028 − 2.15754i 0 0.860404
136.7 0.575085 − 0.256045i 0 −1.07310 + 1.19179i 1.24862 + 0.555922i 0 0.575145 + 0.122251i −0.701028 + 2.15754i 0 0.860404
136.8 1.05062 − 0.467766i 0 −0.453266 + 0.503403i 3.57276 + 1.59069i 0 −4.48724 − 0.953792i −0.951502 + 2.92842i 0 4.49768
136.9 1.24612 − 0.554810i 0 −0.0932499 + 0.103565i −1.89778 − 0.844946i 0 0.101465 + 0.0215671i −0.901773 + 2.77537i 0 −2.83366
136.10 1.67544 − 0.745953i 0 0.912387 − 1.01331i 0.0824312 + 0.0367007i 0 4.60859 + 0.979587i −0.360704 + 1.11013i 0 0.165485
136.11 2.26297 − 1.00754i 0 2.76762 − 3.07376i 2.42411 + 1.07928i 0 −0.234155 − 0.0497713i 1.63516 − 5.03251i 0 6.57309
136.12 2.33604 − 1.04007i 0 3.03706 − 3.37300i −1.88024 − 0.837138i 0 −3.87187 − 0.822992i 2.00615 − 6.17430i 0 −5.26300
190.1 −2.33604 − 1.04007i 0 3.03706 + 3.37300i 1.88024 − 0.837138i 0 −3.87187 + 0.822992i −2.00615 − 6.17430i 0 −5.26300
190.2 −2.26297 − 1.00754i 0 2.76762 + 3.07376i −2.42411 + 1.07928i 0 −0.234155 + 0.0497713i −1.63516 − 5.03251i 0 6.57309
190.3 −1.67544 − 0.745953i 0 0.912387 + 1.01331i −0.0824312 + 0.0367007i 0 4.60859 − 0.979587i 0.360704 + 1.11013i 0 0.165485
190.4 −1.24612 − 0.554810i 0 −0.0932499 − 0.103565i 1.89778 − 0.844946i 0 0.101465 − 0.0215671i 0.901773 + 2.77537i 0 −2.83366
190.5 −1.05062 − 0.467766i 0 −0.453266 − 0.503403i −3.57276 + 1.59069i 0 −4.48724 + 0.953792i 0.951502 + 2.92842i 0 4.49768
190.6 −0.575085 − 0.256045i 0 −1.07310 − 1.19179i −1.24862 + 0.555922i 0 0.575145 − 0.122251i 0.701028 + 2.15754i 0 0.860404
190.7 0.575085 + 0.256045i 0 −1.07310 − 1.19179i 1.24862 − 0.555922i 0 0.575145 − 0.122251i −0.701028 − 2.15754i 0 0.860404
190.8 1.05062 + 0.467766i 0 −0.453266 − 0.503403i 3.57276 − 1.59069i 0 −4.48724 + 0.953792i −0.951502 − 2.92842i 0 4.49768
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner
99.m even 15 1 inner
99.n odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.n.l 96
3.b odd 2 1 inner 891.2.n.l 96
9.c even 3 1 891.2.f.g ✓ 48
9.c even 3 1 inner 891.2.n.l 96
9.d odd 6 1 891.2.f.g ✓ 48
9.d odd 6 1 inner 891.2.n.l 96
11.c even 5 1 inner 891.2.n.l 96
33.h odd 10 1 inner 891.2.n.l 96
99.m even 15 1 891.2.f.g ✓ 48
99.m even 15 1 inner 891.2.n.l 96
99.m even 15 1 9801.2.a.cr 24
99.n odd 30 1 891.2.f.g ✓ 48
99.n odd 30 1 inner 891.2.n.l 96
99.n odd 30 1 9801.2.a.cr 24
99.o odd 30 1 9801.2.a.cq 24
99.p even 30 1 9801.2.a.cq 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
891.2.f.g ✓ 48 9.c even 3 1
891.2.f.g ✓ 48 9.d odd 6 1
891.2.f.g ✓ 48 99.m even 15 1
891.2.f.g ✓ 48 99.n odd 30 1
891.2.n.l 96 1.a even 1 1 trivial
891.2.n.l 96 3.b odd 2 1 inner
891.2.n.l 96 9.c even 3 1 inner
891.2.n.l 96 9.d odd 6 1 inner
891.2.n.l 96 11.c even 5 1 inner
891.2.n.l 96 33.h odd 10 1 inner
891.2.n.l 96 99.m even 15 1 inner
891.2.n.l 96 99.n odd 30 1 inner
9801.2.a.cq 24 99.o odd 30 1
9801.2.a.cq 24 99.p even 30 1
9801.2.a.cr 24 99.m even 15 1
9801.2.a.cr 24 99.n odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 16 T_{2}^{94} + 76 T_{2}^{92} + 556 T_{2}^{90} - 9664 T_{2}^{88} + 60402 T_{2}^{86} + \cdots + 500246412961 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display