Properties

Label 666.2.bj.e
Level $666$
Weight $2$
Character orbit 666.bj
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(289,666)]
 
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, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.bj (of order \(18\), 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 \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 222)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{7} + 6 q^{11} - 6 q^{13} - 18 q^{14} + 30 q^{17} + 24 q^{19} + 12 q^{22} + 24 q^{25} - 12 q^{26} + 6 q^{28} + 12 q^{35} + 24 q^{37} + 36 q^{38} - 6 q^{40} + 48 q^{41} + 12 q^{44} - 12 q^{46} + 6 q^{47} - 18 q^{49} - 72 q^{50} + 12 q^{52} + 30 q^{55} + 6 q^{56} - 36 q^{58} + 48 q^{59} - 6 q^{61} + 12 q^{64} - 54 q^{65} + 36 q^{67} - 48 q^{70} + 36 q^{71} + 12 q^{73} - 18 q^{74} + 24 q^{76} + 30 q^{77} - 30 q^{79} + 18 q^{82} - 18 q^{83} - 36 q^{85} - 42 q^{86} + 18 q^{88} - 6 q^{89} - 48 q^{91} + 30 q^{92} + 30 q^{94} - 42 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 −0.342020 0.939693i 0 −0.766044 + 0.642788i −3.98902 + 0.703371i 0 −0.673690 3.82069i 0.866025 + 0.500000i 0 2.02528 + 3.50788i
289.2 −0.342020 0.939693i 0 −0.766044 + 0.642788i 3.64700 0.643064i 0 −0.469097 2.66038i 0.866025 + 0.500000i 0 −1.85163 3.20712i
289.3 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −1.76204 + 0.310695i 0 −0.343122 1.94594i −0.866025 0.500000i 0 −0.894610 1.54951i
289.4 0.342020 + 0.939693i 0 −0.766044 + 0.642788i 2.10406 0.371002i 0 0.485910 + 2.75573i −0.866025 0.500000i 0 1.06826 + 1.85028i
361.1 −0.984808 + 0.173648i 0 0.939693 0.342020i −0.980233 + 1.16820i 0 −2.89806 2.43176i −0.866025 + 0.500000i 0 0.762486 1.32066i
361.2 −0.984808 + 0.173648i 0 0.939693 0.342020i −0.00457474 + 0.00545196i 0 2.74008 + 2.29920i −0.866025 + 0.500000i 0 0.00355851 0.00616353i
361.3 0.984808 0.173648i 0 0.939693 0.342020i −1.34200 + 1.59933i 0 1.56789 + 1.31561i 0.866025 0.500000i 0 −1.04389 + 1.80807i
361.4 0.984808 0.173648i 0 0.939693 0.342020i 2.32681 2.77298i 0 −2.40991 2.02215i 0.866025 0.500000i 0 1.80993 3.13490i
469.1 −0.642788 0.766044i 0 −0.173648 + 0.984808i −0.743263 2.04210i 0 −2.00421 + 0.729473i 0.866025 0.500000i 0 −1.08658 + 1.88201i
469.2 −0.642788 0.766044i 0 −0.173648 + 0.984808i 0.100476 + 0.276055i 0 2.48902 0.905929i 0.866025 0.500000i 0 0.146886 0.254414i
469.3 0.642788 + 0.766044i 0 −0.173648 + 0.984808i −0.753389 2.06992i 0 −2.61124 + 0.950413i −0.866025 + 0.500000i 0 1.10138 1.90765i
469.4 0.642788 + 0.766044i 0 −0.173648 + 0.984808i 1.39618 + 3.83596i 0 1.12643 0.409987i −0.866025 + 0.500000i 0 −2.04107 + 3.53524i
559.1 −0.984808 0.173648i 0 0.939693 + 0.342020i −0.980233 1.16820i 0 −2.89806 + 2.43176i −0.866025 0.500000i 0 0.762486 + 1.32066i
559.2 −0.984808 0.173648i 0 0.939693 + 0.342020i −0.00457474 0.00545196i 0 2.74008 2.29920i −0.866025 0.500000i 0 0.00355851 + 0.00616353i
559.3 0.984808 + 0.173648i 0 0.939693 + 0.342020i −1.34200 1.59933i 0 1.56789 1.31561i 0.866025 + 0.500000i 0 −1.04389 1.80807i
559.4 0.984808 + 0.173648i 0 0.939693 + 0.342020i 2.32681 + 2.77298i 0 −2.40991 + 2.02215i 0.866025 + 0.500000i 0 1.80993 + 3.13490i
595.1 −0.642788 + 0.766044i 0 −0.173648 0.984808i −0.743263 + 2.04210i 0 −2.00421 0.729473i 0.866025 + 0.500000i 0 −1.08658 1.88201i
595.2 −0.642788 + 0.766044i 0 −0.173648 0.984808i 0.100476 0.276055i 0 2.48902 + 0.905929i 0.866025 + 0.500000i 0 0.146886 + 0.254414i
595.3 0.642788 0.766044i 0 −0.173648 0.984808i −0.753389 + 2.06992i 0 −2.61124 0.950413i −0.866025 0.500000i 0 1.10138 + 1.90765i
595.4 0.642788 0.766044i 0 −0.173648 0.984808i 1.39618 3.83596i 0 1.12643 + 0.409987i −0.866025 0.500000i 0 −2.04107 3.53524i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.bj.e 24
3.b odd 2 1 222.2.n.a 24
37.h even 18 1 inner 666.2.bj.e 24
111.n odd 18 1 222.2.n.a 24
111.q even 36 1 8214.2.a.bh 12
111.q even 36 1 8214.2.a.bj 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.n.a 24 3.b odd 2 1
222.2.n.a 24 111.n odd 18 1
666.2.bj.e 24 1.a even 1 1 trivial
666.2.bj.e 24 37.h even 18 1 inner
8214.2.a.bh 12 111.q even 36 1
8214.2.a.bj 12 111.q even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 12 T_{5}^{22} + 54 T_{5}^{21} - 51 T_{5}^{20} - 882 T_{5}^{19} + 260 T_{5}^{18} + 1998 T_{5}^{17} + 6009 T_{5}^{16} - 49842 T_{5}^{15} + 53868 T_{5}^{14} + 927216 T_{5}^{13} + 2835415 T_{5}^{12} + 1361700 T_{5}^{11} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display