↑

Properties

Label 117.2.f.a
Level $117$
Weight $2$
Character orbit 117.f
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(61,117)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.f (of order \(3\), 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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - q^{3} - 9 q^{4} - 2 q^{5} + 9 q^{6} - 6 q^{7} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - q^{3} - 9 q^{4} - 2 q^{5} + 9 q^{6} - 6 q^{7} - 18 q^{8} - 3 q^{9} - 3 q^{11} - 3 q^{12} + 2 q^{14} + 8 q^{15} - 3 q^{16} + 6 q^{17} - 8 q^{18} - 3 q^{19} + 22 q^{20} - 25 q^{21} + 9 q^{22} - 34 q^{23} - 6 q^{24} - 6 q^{25} - 12 q^{26} + 2 q^{27} + 12 q^{29} + 25 q^{30} - 6 q^{31} + 19 q^{32} - 16 q^{33} + 18 q^{35} + 59 q^{36} - 3 q^{37} + 8 q^{38} - 9 q^{39} - 12 q^{40} - 10 q^{41} - 30 q^{42} + 6 q^{43} + 44 q^{44} - 5 q^{45} - 6 q^{46} + 21 q^{47} - 22 q^{48} - 6 q^{49} + 40 q^{50} + 7 q^{51} - 6 q^{52} - 20 q^{53} - 78 q^{54} + 3 q^{55} - 80 q^{56} + 9 q^{57} - 9 q^{58} - 19 q^{59} + 51 q^{60} + 12 q^{61} + 19 q^{62} - 2 q^{63} - 42 q^{64} + 19 q^{65} - 18 q^{66} + 12 q^{67} - 10 q^{69} + 27 q^{70} + 14 q^{71} + 45 q^{72} + 6 q^{73} - 58 q^{74} - 22 q^{75} + 30 q^{76} + 4 q^{77} + 65 q^{78} + 3 q^{79} - 16 q^{80} + 9 q^{81} - 9 q^{82} - 33 q^{83} + 86 q^{84} + 72 q^{86} + 37 q^{87} + 39 q^{88} - q^{89} + 21 q^{90} - 6 q^{91} - 10 q^{92} + 63 q^{93} + 18 q^{94} + 50 q^{95} - 79 q^{96} - 48 q^{97} - 61 q^{98} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.13984 − 1.97426i −0.286273 + 1.70823i −1.59846 + 2.76861i −1.46964 − 2.54549i 3.69879 − 1.38193i −4.85829 2.72859 −2.83610 − 0.978041i −3.35030 + 5.80289i
61.2 −1.02543 − 1.77610i 1.57384 − 0.723199i −1.10302 + 1.91049i −0.737604 − 1.27757i −2.89834 − 2.05371i 1.16435 0.422561 1.95397 − 2.27640i −1.51272 + 2.62012i
61.3 −0.900808 − 1.56024i −1.30323 + 1.14087i −0.622909 + 1.07891i 1.73153 + 2.99909i 2.95400 + 1.00565i 3.24477 −1.35875 0.396816 − 2.97364i 3.11954 − 5.40321i
61.4 −0.567922 − 0.983670i −0.529547 − 1.64911i 0.354929 − 0.614756i −0.0587384 − 0.101738i −1.32144 + 1.45747i 0.849445 −3.07798 −2.43916 + 1.74657i −0.0667177 + 0.115558i
61.5 −0.348782 − 0.604108i 1.58866 + 0.690034i 0.756702 − 1.31065i 1.44568 + 2.50399i −0.137242 − 1.20040i −3.17282 −2.45082 2.04771 + 2.19246i 1.00846 − 1.74670i
61.6 −0.108182 − 0.187377i 0.455974 + 1.67095i 0.976593 − 1.69151i −0.702153 − 1.21616i 0.263770 − 0.266207i 3.31452 −0.855329 −2.58418 + 1.52382i −0.151921 + 0.263135i
61.7 0.0816825 + 0.141478i −1.72478 + 0.158582i 0.986656 − 1.70894i −1.55806 − 2.69863i −0.163320 − 0.231065i −0.136429 0.649100 2.94970 − 0.547036i 0.254532 − 0.440862i
61.8 0.433689 + 0.751171i 0.753735 − 1.55945i 0.623828 − 1.08050i −0.0324057 − 0.0561283i 1.49830 − 0.110132i −3.92419 2.81694 −1.86377 − 2.35082i 0.0281080 − 0.0486844i
61.9 0.602131 + 1.04292i −1.54217 − 0.788485i 0.274876 − 0.476100i 1.89177 + 3.27665i −0.106261 − 2.08313i 0.300456 3.07057 1.75658 + 2.43196i −2.27819 + 3.94594i
61.10 1.00395 + 1.73890i 1.56525 + 0.741609i −1.01584 + 1.75949i −1.37329 − 2.37860i 0.281858 + 3.46635i −2.23810 −0.0636126 1.90003 + 2.32161i 2.75743 − 4.77600i
61.11 1.14137 + 1.97690i −1.25251 + 1.19634i −1.60543 + 2.78069i −0.461458 − 0.799268i −3.79461 − 1.11063i 0.908370 −2.76408 0.137559 − 2.99684i 1.05338 − 1.82452i
61.12 1.32814 + 2.30041i 0.201038 − 1.72034i −2.52792 + 4.37849i 0.324360 + 0.561808i 4.22450 − 1.82239i 1.54792 −8.11720 −2.91917 − 0.691710i −0.861592 + 1.49232i
94.1 −1.13984 + 1.97426i −0.286273 − 1.70823i −1.59846 − 2.76861i −1.46964 + 2.54549i 3.69879 + 1.38193i −4.85829 2.72859 −2.83610 + 0.978041i −3.35030 − 5.80289i
94.2 −1.02543 + 1.77610i 1.57384 + 0.723199i −1.10302 − 1.91049i −0.737604 + 1.27757i −2.89834 + 2.05371i 1.16435 0.422561 1.95397 + 2.27640i −1.51272 − 2.62012i
94.3 −0.900808 + 1.56024i −1.30323 − 1.14087i −0.622909 − 1.07891i 1.73153 − 2.99909i 2.95400 − 1.00565i 3.24477 −1.35875 0.396816 + 2.97364i 3.11954 + 5.40321i
94.4 −0.567922 + 0.983670i −0.529547 + 1.64911i 0.354929 + 0.614756i −0.0587384 + 0.101738i −1.32144 − 1.45747i 0.849445 −3.07798 −2.43916 − 1.74657i −0.0667177 − 0.115558i
94.5 −0.348782 + 0.604108i 1.58866 − 0.690034i 0.756702 + 1.31065i 1.44568 − 2.50399i −0.137242 + 1.20040i −3.17282 −2.45082 2.04771 − 2.19246i 1.00846 + 1.74670i
94.6 −0.108182 + 0.187377i 0.455974 − 1.67095i 0.976593 + 1.69151i −0.702153 + 1.21616i 0.263770 + 0.266207i 3.31452 −0.855329 −2.58418 − 1.52382i −0.151921 − 0.263135i
94.7 0.0816825 − 0.141478i −1.72478 − 0.158582i 0.986656 + 1.70894i −1.55806 + 2.69863i −0.163320 + 0.231065i −0.136429 0.649100 2.94970 + 0.547036i 0.254532 + 0.440862i
94.8 0.433689 − 0.751171i 0.753735 + 1.55945i 0.623828 + 1.08050i −0.0324057 + 0.0561283i 1.49830 + 0.110132i −3.92419 2.81694 −1.86377 + 2.35082i 0.0281080 + 0.0486844i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.f.a ✓ 24
3.b odd 2 1 351.2.f.a 24
9.c even 3 1 117.2.h.a yes 24
9.d odd 6 1 351.2.h.a 24
13.c even 3 1 117.2.h.a yes 24
39.i odd 6 1 351.2.h.a 24
117.f even 3 1 inner 117.2.f.a ✓ 24
117.u odd 6 1 351.2.f.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.f.a ✓ 24 1.a even 1 1 trivial
117.2.f.a ✓ 24 117.f even 3 1 inner
117.2.h.a yes 24 9.c even 3 1
117.2.h.a yes 24 13.c even 3 1
351.2.f.a 24 3.b odd 2 1
351.2.f.a 24 117.u odd 6 1
351.2.h.a 24 9.d odd 6 1
351.2.h.a 24 39.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(117, [\chi])\).