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Properties

Label 117.2.r.b
Level $117$
Weight $2$
Character orbit 117.r
Analytic conductor $0.934$
Analytic rank $0$
Dimension $22$
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] = [117,2,Mod(43,117)]
 
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([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.r (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} + 10 q^{4} + 3 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} + 10 q^{4} + 3 q^{5} - 6 q^{6} - 2 q^{9} - 7 q^{10} + 3 q^{11} - 11 q^{12} + 3 q^{13} - 9 q^{14} - 15 q^{15} - 12 q^{16} + 9 q^{17} - 30 q^{18} - 6 q^{19} + 12 q^{21} - 13 q^{22} - 12 q^{23} + 45 q^{24} + 4 q^{25} - 12 q^{26} + 14 q^{27} + 3 q^{28} - 24 q^{29} - 39 q^{30} + 27 q^{31} - 18 q^{33} - 15 q^{34} - 27 q^{35} + 2 q^{36} + 6 q^{37} + 21 q^{38} + 13 q^{39} + 13 q^{40} + 27 q^{42} + 8 q^{43} + 27 q^{45} - 15 q^{46} + 6 q^{47} + 11 q^{48} - 14 q^{49} + 12 q^{51} - 7 q^{52} - 24 q^{53} - 33 q^{54} - 13 q^{55} + 18 q^{56} + 42 q^{57} + 15 q^{58} - 33 q^{59} - 6 q^{60} - 6 q^{61} - 3 q^{63} - 24 q^{64} + 3 q^{65} + 30 q^{66} + 138 q^{68} - 3 q^{69} + 24 q^{70} + 9 q^{71} - 6 q^{72} + 12 q^{74} + 22 q^{75} + 42 q^{77} - 42 q^{78} - 6 q^{79} + 105 q^{80} + 10 q^{81} - 16 q^{82} - 42 q^{83} - 18 q^{84} - 51 q^{85} - 45 q^{86} + 27 q^{87} - 11 q^{88} - 30 q^{89} + 30 q^{90} + 15 q^{91} - 3 q^{92} + 6 q^{93} - 88 q^{94} - 3 q^{95} + 9 q^{96} + 117 q^{98} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −2.24331 + 1.29518i −0.749892 − 1.56130i 2.35496 − 4.07892i −1.18696 + 0.685292i 3.70440 + 2.53124i 3.70457i 7.01967i −1.87532 + 2.34162i 1.77515 − 3.07465i
43.2 −2.00627 + 1.15832i 1.71454 − 0.245682i 1.68341 − 2.91575i 1.09505 − 0.632228i −3.15524 + 2.47889i − 4.17527i 3.16642i 2.87928 − 0.842464i −1.46464 + 2.53684i
43.3 −1.21740 + 0.702869i 0.712485 + 1.57872i −0.0119503 + 0.0206986i 2.61504 − 1.50979i −1.97702 − 1.42116i 3.19463i − 2.84507i −1.98473 + 2.24963i −2.12237 + 3.67605i
43.4 −0.916018 + 0.528863i −1.42564 − 0.983643i −0.440607 + 0.763154i 2.71101 − 1.56520i 1.82612 + 0.147067i − 0.906314i − 3.04754i 1.06489 + 2.80464i −1.65555 + 2.86750i
43.5 −0.838455 + 0.484082i 1.72848 − 0.111177i −0.531329 + 0.920289i −3.54737 + 2.04808i −1.39543 + 0.929943i 3.54220i − 2.96516i 2.97528 − 0.384334i 1.98287 − 3.43444i
43.6 0.339230 − 0.195855i 1.10027 − 1.33768i −0.923282 + 1.59917i 1.60580 − 0.927107i 0.111254 − 0.669277i 0.0822579i 1.50674i −0.578797 − 2.94364i 0.363157 − 0.629006i
43.7 0.495326 − 0.285977i −1.06025 + 1.36962i −0.836435 + 1.44875i −0.796103 + 0.459630i −0.133491 + 0.981617i 1.93281i 2.10071i −0.751725 − 2.90429i −0.262887 + 0.455333i
43.8 0.677814 − 0.391336i 1.22535 + 1.22414i −0.693712 + 1.20154i −0.0536139 + 0.0309540i 1.30961 + 0.350219i − 3.75567i 2.65124i 0.00294547 + 3.00000i −0.0242268 + 0.0419621i
43.9 1.67544 − 0.967314i −1.73126 + 0.0524448i 0.871392 − 1.50930i 2.26677 − 1.30872i −2.84988 + 1.76254i − 2.32894i 0.497616i 2.99450 − 0.181591i 2.53189 − 4.38536i
43.10 1.73739 − 1.00309i −0.305519 − 1.70489i 1.01236 − 1.75346i −0.778411 + 0.449416i −2.24096 − 2.65561i 2.43501i − 0.0495935i −2.81332 + 1.04175i −0.901605 + 1.56163i
43.11 2.29626 − 1.32574i −0.208561 + 1.71945i 2.51519 − 4.35644i −2.43120 + 1.40366i 1.80064 + 4.22479i − 0.261179i − 8.03502i −2.91300 − 0.717221i −3.72178 + 6.44631i
49.1 −2.24331 − 1.29518i −0.749892 + 1.56130i 2.35496 + 4.07892i −1.18696 − 0.685292i 3.70440 − 2.53124i − 3.70457i − 7.01967i −1.87532 − 2.34162i 1.77515 + 3.07465i
49.2 −2.00627 − 1.15832i 1.71454 + 0.245682i 1.68341 + 2.91575i 1.09505 + 0.632228i −3.15524 − 2.47889i 4.17527i − 3.16642i 2.87928 + 0.842464i −1.46464 − 2.53684i
49.3 −1.21740 − 0.702869i 0.712485 − 1.57872i −0.0119503 − 0.0206986i 2.61504 + 1.50979i −1.97702 + 1.42116i − 3.19463i 2.84507i −1.98473 − 2.24963i −2.12237 − 3.67605i
49.4 −0.916018 − 0.528863i −1.42564 + 0.983643i −0.440607 − 0.763154i 2.71101 + 1.56520i 1.82612 − 0.147067i 0.906314i 3.04754i 1.06489 − 2.80464i −1.65555 − 2.86750i
49.5 −0.838455 − 0.484082i 1.72848 + 0.111177i −0.531329 − 0.920289i −3.54737 − 2.04808i −1.39543 − 0.929943i − 3.54220i 2.96516i 2.97528 + 0.384334i 1.98287 + 3.43444i
49.6 0.339230 + 0.195855i 1.10027 + 1.33768i −0.923282 − 1.59917i 1.60580 + 0.927107i 0.111254 + 0.669277i − 0.0822579i − 1.50674i −0.578797 + 2.94364i 0.363157 + 0.629006i
49.7 0.495326 + 0.285977i −1.06025 − 1.36962i −0.836435 − 1.44875i −0.796103 − 0.459630i −0.133491 − 0.981617i − 1.93281i − 2.10071i −0.751725 + 2.90429i −0.262887 − 0.455333i
49.8 0.677814 + 0.391336i 1.22535 − 1.22414i −0.693712 − 1.20154i −0.0536139 − 0.0309540i 1.30961 − 0.350219i 3.75567i − 2.65124i 0.00294547 − 3.00000i −0.0242268 − 0.0419621i
49.9 1.67544 + 0.967314i −1.73126 − 0.0524448i 0.871392 + 1.50930i 2.26677 + 1.30872i −2.84988 − 1.76254i 2.32894i − 0.497616i 2.99450 + 0.181591i 2.53189 + 4.38536i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.11
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 16 T_{2}^{20} + 168 T_{2}^{18} - 1012 T_{2}^{16} + 4402 T_{2}^{14} - 11910 T_{2}^{12} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display