Properties

Label 117.2.l.b
Level $117$
Weight $2$
Character orbit 117.l
Analytic conductor $0.934$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(4,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.l (of order \(6\), degree \(2\), 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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{3} - 20 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{3} - 20 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} + 7 q^{9} - 7 q^{10} - 11 q^{12} - 9 q^{14} - 6 q^{15} + 24 q^{16} + 9 q^{17} + 30 q^{18} - 6 q^{19} - 24 q^{20} - 12 q^{21} + 26 q^{22} + 6 q^{23} + 24 q^{24} + 4 q^{25} - 12 q^{26} + 14 q^{27} + 3 q^{28} + 48 q^{29} + 9 q^{30} - 27 q^{31} - 3 q^{33} + 15 q^{34} - 27 q^{35} - 34 q^{36} + 6 q^{37} + 21 q^{38} + 19 q^{39} + 13 q^{40} + 6 q^{41} - 30 q^{42} - 4 q^{43} + 6 q^{45} - 15 q^{46} - 6 q^{47} - 13 q^{48} + 7 q^{49} + 18 q^{50} + 12 q^{51} - 22 q^{52} - 24 q^{53} + 21 q^{54} - 13 q^{55} - 9 q^{56} - 42 q^{57} + 6 q^{60} + 3 q^{61} - 54 q^{63} - 24 q^{64} - 57 q^{65} + 30 q^{66} - 45 q^{67} - 69 q^{68} + 42 q^{69} - 24 q^{70} + 9 q^{71} - 24 q^{72} - 6 q^{74} - 8 q^{75} + 18 q^{76} + 42 q^{77} + 6 q^{78} - 6 q^{79} + 105 q^{80} + 19 q^{81} - 16 q^{82} + 42 q^{83} + 66 q^{84} + 45 q^{86} - 6 q^{87} + 22 q^{88} - 30 q^{89} + 30 q^{90} + 15 q^{91} - 3 q^{92} + 39 q^{93} + 44 q^{94} + 6 q^{95} - 9 q^{96} - 27 q^{97} + 117 q^{98} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 2.59035i 1.72707 0.131225i −4.70993 1.18696 + 0.685292i −0.339918 4.47373i −3.20825 1.85228i 7.01967i 2.96556 0.453269i 1.77515 3.07465i
4.2 2.31664i −0.644502 1.60767i −3.36682 −1.09505 0.632228i −3.72440 + 1.49308i 3.61589 + 2.08764i 3.16642i −2.16923 + 2.07230i −1.46464 + 2.53684i
4.3 1.40574i −1.72346 + 0.172331i 0.0239006 −2.61504 1.50979i 0.242252 + 2.42273i −2.76663 1.59731i 2.84507i 2.94060 0.594010i −2.12237 + 3.67605i
4.4 1.05773i 1.56468 + 0.742818i 0.881215 −2.71101 1.56520i 0.785698 1.65500i 0.784891 + 0.453157i 3.04754i 1.89644 + 2.32454i −1.65555 + 2.86750i
4.5 0.968164i −0.767957 1.55250i 1.06266 3.54737 + 2.04808i −1.50307 + 0.743509i −3.06763 1.77110i 2.96516i −1.82048 + 2.38450i 1.98287 3.43444i
4.6 0.391710i 0.608332 1.62171i 1.84656 −1.60580 0.927107i 0.635238 + 0.238289i −0.0712374 0.0411289i 1.50674i −2.25987 1.97307i 0.363157 0.629006i
4.7 0.571953i −0.656000 + 1.60302i 1.67287 0.796103 + 0.459630i −0.916851 0.375201i −1.67386 0.966405i 2.10071i −2.13933 2.10316i −0.262887 + 0.455333i
4.8 0.782672i −1.67281 0.449109i 1.38742 0.0536139 + 0.0309540i 0.351505 1.30926i 3.25250 + 1.87783i 2.65124i 2.59660 + 1.50255i −0.0242268 + 0.0419621i
4.9 1.93463i 0.820210 + 1.52553i −1.74278 −2.26677 1.30872i −2.95134 + 1.58680i 2.01692 + 1.16447i 0.497616i −1.65451 + 2.50252i 2.53189 4.38536i
4.10 2.00617i 1.62924 0.587859i −2.02472 0.778411 + 0.449416i 1.17935 + 3.26853i −2.10878 1.21751i 0.0495935i 2.30884 1.91553i −0.901605 + 1.56163i
4.11 2.65149i −1.38481 + 1.04034i −5.03038 2.43120 + 1.40366i −2.75846 3.67179i 0.226187 + 0.130589i 8.03502i 0.835370 2.88135i −3.72178 + 6.44631i
88.1 2.65149i −1.38481 1.04034i −5.03038 2.43120 1.40366i −2.75846 + 3.67179i 0.226187 0.130589i 8.03502i 0.835370 + 2.88135i −3.72178 6.44631i
88.2 2.00617i 1.62924 + 0.587859i −2.02472 0.778411 0.449416i 1.17935 3.26853i −2.10878 + 1.21751i 0.0495935i 2.30884 + 1.91553i −0.901605 1.56163i
88.3 1.93463i 0.820210 1.52553i −1.74278 −2.26677 + 1.30872i −2.95134 1.58680i 2.01692 1.16447i 0.497616i −1.65451 2.50252i 2.53189 + 4.38536i
88.4 0.782672i −1.67281 + 0.449109i 1.38742 0.0536139 0.0309540i 0.351505 + 1.30926i 3.25250 1.87783i 2.65124i 2.59660 1.50255i −0.0242268 0.0419621i
88.5 0.571953i −0.656000 1.60302i 1.67287 0.796103 0.459630i −0.916851 + 0.375201i −1.67386 + 0.966405i 2.10071i −2.13933 + 2.10316i −0.262887 0.455333i
88.6 0.391710i 0.608332 + 1.62171i 1.84656 −1.60580 + 0.927107i 0.635238 0.238289i −0.0712374 + 0.0411289i 1.50674i −2.25987 + 1.97307i 0.363157 + 0.629006i
88.7 0.968164i −0.767957 + 1.55250i 1.06266 3.54737 2.04808i −1.50307 0.743509i −3.06763 + 1.77110i 2.96516i −1.82048 2.38450i 1.98287 + 3.43444i
88.8 1.05773i 1.56468 0.742818i 0.881215 −2.71101 + 1.56520i 0.785698 + 1.65500i 0.784891 0.453157i 3.04754i 1.89644 2.32454i −1.65555 2.86750i
88.9 1.40574i −1.72346 0.172331i 0.0239006 −2.61504 + 1.50979i 0.242252 2.42273i −2.76663 + 1.59731i 2.84507i 2.94060 + 0.594010i −2.12237 3.67605i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.l.b 22
3.b odd 2 1 351.2.l.b 22
9.c even 3 1 117.2.r.b yes 22
9.d odd 6 1 351.2.r.b 22
13.e even 6 1 117.2.r.b yes 22
39.h odd 6 1 351.2.r.b 22
117.l even 6 1 inner 117.2.l.b 22
117.v odd 6 1 351.2.l.b 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.l.b 22 1.a even 1 1 trivial
117.2.l.b 22 117.l even 6 1 inner
117.2.r.b yes 22 9.c even 3 1
117.2.r.b yes 22 13.e even 6 1
351.2.l.b 22 3.b odd 2 1
351.2.l.b 22 117.v odd 6 1
351.2.r.b 22 9.d odd 6 1
351.2.r.b 22 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 32 T_{2}^{20} + 432 T_{2}^{18} + 3212 T_{2}^{16} + 14428 T_{2}^{14} + 40524 T_{2}^{12} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display