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Properties

Label 666.2.x.i
Level $666$
Weight $2$
Character orbit 666.x
Analytic conductor $5.318$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(127,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.x (of order \(9\), degree \(6\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{8} - 6 q^{10} + 6 q^{11} + 12 q^{17} + 6 q^{19} + 24 q^{23} - 12 q^{25} + 12 q^{29} - 12 q^{31} - 12 q^{34} - 6 q^{35} - 6 q^{37} + 24 q^{38} + 36 q^{41} - 24 q^{43} - 18 q^{46} - 12 q^{47} - 6 q^{49} - 6 q^{50} + 24 q^{53} + 36 q^{55} - 12 q^{58} + 42 q^{61} - 12 q^{62} - 12 q^{64} - 30 q^{65} + 12 q^{67} - 24 q^{68} + 6 q^{70} - 6 q^{71} + 36 q^{73} + 6 q^{74} + 6 q^{76} + 60 q^{77} + 24 q^{79} - 12 q^{80} + 12 q^{82} + 12 q^{83} - 24 q^{85} + 6 q^{86} - 6 q^{88} - 36 q^{89} + 72 q^{91} - 6 q^{94} - 132 q^{95} - 54 q^{97} + 42 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.685182 + 3.88586i 0 −0.238222 + 1.35102i 0.500000 + 0.866025i 0 −1.97290 + 3.41717i
127.2 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.255073 + 1.44659i 0 0.767063 − 4.35023i 0.500000 + 0.866025i 0 −0.734452 + 1.27211i
127.3 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.0551662 + 0.312863i 0 −0.563253 + 3.19437i 0.500000 + 0.866025i 0 −0.158845 + 0.275127i
127.4 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.648124 − 3.67569i 0 0.0344122 − 0.195161i 0.500000 + 0.866025i 0 1.86620 − 3.23235i
145.1 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −2.30535 − 0.839077i 0 1.96257 + 0.714317i 0.500000 − 0.866025i 0 1.22665 + 2.12462i
145.2 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −1.01783 − 0.370460i 0 −0.173626 − 0.0631945i 0.500000 − 0.866025i 0 0.541576 + 0.938038i
145.3 −0.766044 − 0.642788i 0 0.173648 + 0.984808i 1.56809 + 0.570738i 0 −3.64428 − 1.32641i 0.500000 − 0.866025i 0 −0.834363 − 1.44516i
145.4 −0.766044 − 0.642788i 0 0.173648 + 0.984808i 3.63447 + 1.32284i 0 1.85534 + 0.675287i 0.500000 − 0.866025i 0 −1.93386 − 3.34955i
181.1 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i −2.46730 + 2.07031i 0 3.54159 − 2.97174i 0.500000 − 0.866025i 0 −1.61041 − 2.78932i
181.2 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i −1.96856 + 1.65182i 0 −3.79061 + 3.18070i 0.500000 − 0.866025i 0 −1.28488 − 2.22549i
181.3 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 0.491477 − 0.412398i 0 −1.28775 + 1.08055i 0.500000 − 0.866025i 0 0.320789 + 0.555622i
181.4 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 2.41229 − 2.02415i 0 1.53678 − 1.28951i 0.500000 − 0.866025i 0 1.57451 + 2.72713i
271.1 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −2.30535 + 0.839077i 0 1.96257 − 0.714317i 0.500000 + 0.866025i 0 1.22665 − 2.12462i
271.2 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −1.01783 + 0.370460i 0 −0.173626 + 0.0631945i 0.500000 + 0.866025i 0 0.541576 − 0.938038i
271.3 −0.766044 + 0.642788i 0 0.173648 − 0.984808i 1.56809 − 0.570738i 0 −3.64428 + 1.32641i 0.500000 + 0.866025i 0 −0.834363 + 1.44516i
271.4 −0.766044 + 0.642788i 0 0.173648 − 0.984808i 3.63447 − 1.32284i 0 1.85534 − 0.675287i 0.500000 + 0.866025i 0 −1.93386 + 3.34955i
379.1 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i −2.46730 − 2.07031i 0 3.54159 + 2.97174i 0.500000 + 0.866025i 0 −1.61041 + 2.78932i
379.2 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i −1.96856 − 1.65182i 0 −3.79061 − 3.18070i 0.500000 + 0.866025i 0 −1.28488 + 2.22549i
379.3 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i 0.491477 + 0.412398i 0 −1.28775 − 1.08055i 0.500000 + 0.866025i 0 0.320789 − 0.555622i
379.4 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i 2.41229 + 2.02415i 0 1.53678 + 1.28951i 0.500000 + 0.866025i 0 1.57451 − 2.72713i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.x.i yes 24
3.b odd 2 1 666.2.x.h ✓ 24
37.f even 9 1 inner 666.2.x.i yes 24
111.p odd 18 1 666.2.x.h ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.x.h ✓ 24 3.b odd 2 1
666.2.x.h ✓ 24 111.p odd 18 1
666.2.x.i yes 24 1.a even 1 1 trivial
666.2.x.i yes 24 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 6 T_{5}^{22} - 26 T_{5}^{21} - 57 T_{5}^{20} - 84 T_{5}^{19} + 2000 T_{5}^{18} + \cdots + 3884841 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display