Properties

Label 990.2.bh.c
Level $990$
Weight $2$
Character orbit 990.bh
Analytic conductor $7.905$
Analytic rank $0$
Dimension $48$
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] = [990,2,Mod(73,990)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(990, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 15, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("990.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.bh (of order \(20\), degree \(8\), 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: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{5} - 20 q^{7} - 12 q^{11} + 12 q^{16} + 20 q^{17} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 20 q^{25} - 8 q^{26} - 20 q^{28} + 16 q^{31} + 20 q^{37} + 36 q^{38} + 20 q^{41} + 40 q^{46} - 40 q^{47} - 40 q^{50} + 40 q^{52} + 96 q^{55} + 8 q^{56} + 48 q^{58} + 80 q^{61} - 40 q^{62} - 20 q^{68} - 56 q^{70} + 56 q^{71} - 20 q^{73} + 96 q^{77} + 12 q^{80} - 80 q^{85} + 56 q^{86} - 4 q^{88} - 68 q^{91} + 12 q^{92} - 20 q^{95} + 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.156434 0.987688i 0 −0.951057 + 0.309017i −2.14909 0.617590i 0 −0.00697389 + 0.0136870i 0.453990 + 0.891007i 0 −0.273795 + 2.21924i
73.2 −0.156434 0.987688i 0 −0.951057 + 0.309017i 1.90161 1.17638i 0 −0.692077 + 1.35828i 0.453990 + 0.891007i 0 −1.45937 1.69418i
73.3 −0.156434 0.987688i 0 −0.951057 + 0.309017i 2.03838 + 0.919234i 0 −1.55752 + 3.05681i 0.453990 + 0.891007i 0 0.589044 2.15709i
73.4 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −1.55051 1.61119i 0 −1.46937 + 2.88379i −0.453990 0.891007i 0 1.34880 1.78346i
73.5 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −0.923903 2.03627i 0 1.49255 2.92930i −0.453990 0.891007i 0 1.86667 1.23107i
73.6 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.507931 + 2.17761i 0 −1.71860 + 3.37294i −0.453990 0.891007i 0 −2.07135 + 0.842332i
127.1 −0.891007 0.453990i 0 0.587785 + 0.809017i −1.60893 + 1.55285i 0 −0.0165943 + 0.104772i −0.156434 0.987688i 0 2.13855 0.653156i
127.2 −0.891007 0.453990i 0 0.587785 + 0.809017i −0.537022 2.17062i 0 −0.516545 + 3.26134i −0.156434 0.987688i 0 −0.506953 + 2.17784i
127.3 −0.891007 0.453990i 0 0.587785 + 0.809017i 2.22860 0.182641i 0 0.119289 0.753160i −0.156434 0.987688i 0 −2.06861 0.849027i
127.4 0.891007 + 0.453990i 0 0.587785 + 0.809017i −2.18748 + 0.463629i 0 0.620489 3.91761i 0.156434 + 0.987688i 0 −2.15954 0.579997i
127.5 0.891007 + 0.453990i 0 0.587785 + 0.809017i −0.667967 2.13397i 0 −0.0611439 + 0.386047i 0.156434 + 0.987688i 0 0.373639 2.20463i
127.6 0.891007 + 0.453990i 0 0.587785 + 0.809017i 1.87069 1.22496i 0 −0.466963 + 2.94829i 0.156434 + 0.987688i 0 2.22292 0.242173i
217.1 −0.156434 + 0.987688i 0 −0.951057 0.309017i −2.14909 + 0.617590i 0 −0.00697389 0.0136870i 0.453990 0.891007i 0 −0.273795 2.21924i
217.2 −0.156434 + 0.987688i 0 −0.951057 0.309017i 1.90161 + 1.17638i 0 −0.692077 1.35828i 0.453990 0.891007i 0 −1.45937 + 1.69418i
217.3 −0.156434 + 0.987688i 0 −0.951057 0.309017i 2.03838 0.919234i 0 −1.55752 3.05681i 0.453990 0.891007i 0 0.589044 + 2.15709i
217.4 0.156434 0.987688i 0 −0.951057 0.309017i −1.55051 + 1.61119i 0 −1.46937 2.88379i −0.453990 + 0.891007i 0 1.34880 + 1.78346i
217.5 0.156434 0.987688i 0 −0.951057 0.309017i −0.923903 + 2.03627i 0 1.49255 + 2.92930i −0.453990 + 0.891007i 0 1.86667 + 1.23107i
217.6 0.156434 0.987688i 0 −0.951057 0.309017i 0.507931 2.17761i 0 −1.71860 3.37294i −0.453990 + 0.891007i 0 −2.07135 0.842332i
343.1 −0.891007 + 0.453990i 0 0.587785 0.809017i −1.60893 1.55285i 0 −0.0165943 0.104772i −0.156434 + 0.987688i 0 2.13855 + 0.653156i
343.2 −0.891007 + 0.453990i 0 0.587785 0.809017i −0.537022 + 2.17062i 0 −0.516545 3.26134i −0.156434 + 0.987688i 0 −0.506953 2.17784i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.bh.c 48
3.b odd 2 1 110.2.k.a 48
5.c odd 4 1 inner 990.2.bh.c 48
11.d odd 10 1 inner 990.2.bh.c 48
12.b even 2 1 880.2.cm.c 48
15.d odd 2 1 550.2.bh.b 48
15.e even 4 1 110.2.k.a 48
15.e even 4 1 550.2.bh.b 48
33.f even 10 1 110.2.k.a 48
55.l even 20 1 inner 990.2.bh.c 48
60.l odd 4 1 880.2.cm.c 48
132.n odd 10 1 880.2.cm.c 48
165.r even 10 1 550.2.bh.b 48
165.u odd 20 1 110.2.k.a 48
165.u odd 20 1 550.2.bh.b 48
660.bv even 20 1 880.2.cm.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.k.a 48 3.b odd 2 1
110.2.k.a 48 15.e even 4 1
110.2.k.a 48 33.f even 10 1
110.2.k.a 48 165.u odd 20 1
550.2.bh.b 48 15.d odd 2 1
550.2.bh.b 48 15.e even 4 1
550.2.bh.b 48 165.r even 10 1
550.2.bh.b 48 165.u odd 20 1
880.2.cm.c 48 12.b even 2 1
880.2.cm.c 48 60.l odd 4 1
880.2.cm.c 48 132.n odd 10 1
880.2.cm.c 48 660.bv even 20 1
990.2.bh.c 48 1.a even 1 1 trivial
990.2.bh.c 48 5.c odd 4 1 inner
990.2.bh.c 48 11.d odd 10 1 inner
990.2.bh.c 48 55.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} + 20 T_{7}^{47} + 200 T_{7}^{46} + 1280 T_{7}^{45} + 5417 T_{7}^{44} + 12340 T_{7}^{43} - 17400 T_{7}^{42} - 307780 T_{7}^{41} - 1496254 T_{7}^{40} - 3777740 T_{7}^{39} + 131200 T_{7}^{38} + 47252320 T_{7}^{37} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\). Copy content Toggle raw display