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Properties

Label 117.3.u.a
Level $117$
Weight $3$
Character orbit 117.u
Analytic conductor $3.188$
Analytic rank $0$
Dimension $52$
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] = [117,3,Mod(68,117)]
 
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([5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.68");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.u (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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 3 q^{2} - q^{3} + 49 q^{4} - 6 q^{5} - 3 q^{6} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 3 q^{2} - q^{3} + 49 q^{4} - 6 q^{5} - 3 q^{6} + 2 q^{7} - 3 q^{9} - 6 q^{10} + 33 q^{11} - 39 q^{12} + 4 q^{13} - 6 q^{14} - 28 q^{15} - 83 q^{16} + 34 q^{18} + 5 q^{19} - 91 q^{21} - 15 q^{22} + 36 q^{24} + 88 q^{25} + 132 q^{26} - 34 q^{27} - 22 q^{28} - 30 q^{29} + 31 q^{30} + 14 q^{31} - 63 q^{32} - 112 q^{33} - 6 q^{34} - 228 q^{35} + 47 q^{36} - 13 q^{37} - 192 q^{38} + 147 q^{39} + 72 q^{40} - 84 q^{42} + 62 q^{43} - 107 q^{45} + 6 q^{46} - 69 q^{47} - 58 q^{48} + 246 q^{49} + 235 q^{51} + 112 q^{52} + 444 q^{54} - 27 q^{55} + 189 q^{57} - 105 q^{58} - 435 q^{59} + 213 q^{60} - 16 q^{61} - 471 q^{62} - 170 q^{63} - 98 q^{64} - 435 q^{65} - 576 q^{66} + 68 q^{67} + 152 q^{69} - 57 q^{70} - 144 q^{71} + 513 q^{72} - 106 q^{73} + 98 q^{75} + 158 q^{76} + 282 q^{77} - 433 q^{78} - 25 q^{79} + 1050 q^{80} + 9 q^{81} + 81 q^{82} + 219 q^{83} - 172 q^{84} + 54 q^{85} - 24 q^{86} + 325 q^{87} - 15 q^{88} - 99 q^{89} + 183 q^{90} + 2 q^{91} + 888 q^{92} - 171 q^{93} - 606 q^{94} - 258 q^{95} - 439 q^{96} - 424 q^{97} + 405 q^{98} + 522 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −3.34685 − 1.93231i −2.94217 + 0.586194i 5.46761 + 9.47019i 0.468040 + 0.270223i 10.9797 + 3.72327i 0.553915 − 26.8020i 8.31275 − 3.44937i −1.04431 − 1.80879i
68.2 −3.06533 − 1.76977i 2.49553 + 1.66503i 4.26415 + 7.38572i −6.00755 − 3.46846i −4.70290 − 9.52037i 4.78124 − 16.0281i 3.45535 + 8.31027i 12.2767 + 21.2639i
68.3 −2.83378 − 1.63609i 0.158993 − 2.99578i 3.35355 + 5.80853i −3.48512 − 2.01214i −5.35191 + 8.22928i −2.80059 − 8.85812i −8.94944 − 0.952618i 6.58406 + 11.4039i
68.4 −2.67828 − 1.54631i 1.65993 + 2.49893i 2.78213 + 4.81880i 6.70477 + 3.87100i −0.581647 − 9.25959i −12.6481 − 4.83767i −3.48928 + 8.29608i −11.9715 − 20.7353i
68.5 −2.39019 − 1.37998i −1.29328 − 2.70692i 1.80866 + 3.13270i 7.95372 + 4.59208i −0.644309 + 8.25474i 5.43975 1.05616i −5.65486 + 7.00161i −12.6739 − 21.9519i
68.6 −2.23352 − 1.28952i −0.951185 + 2.84521i 1.32573 + 2.29623i −0.817198 − 0.471810i 5.79345 − 5.12826i 5.04701 3.47795i −7.19049 − 5.41265i 1.21682 + 2.10759i
68.7 −2.11714 − 1.22233i 2.89505 − 0.786578i 0.988182 + 1.71158i 2.73246 + 1.57759i −7.09067 − 1.87341i 7.65083 4.94710i 7.76259 − 4.55436i −3.85666 − 6.67994i
68.8 −1.61282 − 0.931162i −2.77624 + 1.13689i −0.265873 − 0.460506i 0.895843 + 0.517215i 5.53620 + 0.751532i −5.11682 8.43958i 6.41497 − 6.31253i −0.963222 − 1.66835i
68.9 −1.46109 − 0.843560i 2.47307 − 1.69821i −0.576814 − 0.999071i −3.08450 − 1.78084i −5.04592 + 0.395052i −12.7968 8.69479i 3.23216 − 8.39959i 3.00448 + 5.20392i
68.10 −1.44683 − 0.835327i −2.64561 − 1.41449i −0.604458 − 1.04695i −8.08978 − 4.67064i 2.64618 + 4.25647i 8.35713 8.70230i 4.99845 + 7.48435i 7.80302 + 13.5152i
68.11 −0.545922 − 0.315188i 0.359457 + 2.97839i −1.80131 − 3.11997i 1.96357 + 1.13367i 0.742517 − 1.73926i 8.19462 4.79251i −8.74158 + 2.14120i −0.714636 − 1.23779i
68.12 −0.422456 − 0.243905i 1.49896 + 2.59867i −1.88102 − 3.25802i −6.04718 − 3.49134i 0.000583198 − 1.46343i −7.27212 3.78640i −4.50621 + 7.79064i 1.70311 + 2.94987i
68.13 −0.376626 − 0.217445i −2.32751 − 1.89280i −1.90544 − 3.30031i 3.11857 + 1.80051i 0.465021 + 1.21898i −7.89006 3.39687i 1.83461 + 8.81103i −0.783024 − 1.35624i
68.14 −0.325651 − 0.188014i 0.894719 − 2.86347i −1.92930 − 3.34165i 0.994641 + 0.574256i −0.829740 + 0.764272i 5.14423 2.95506i −7.39896 − 5.12401i −0.215937 − 0.374014i
68.15 0.265125 + 0.153070i 2.88266 + 0.830829i −1.95314 − 3.38294i 5.23834 + 3.02436i 0.637090 + 0.661523i −0.615739 − 2.42043i 7.61944 + 4.79000i 0.925877 + 1.60367i
68.16 0.979822 + 0.565700i 2.79207 − 1.09742i −1.35997 − 2.35553i −7.06649 − 4.07984i 3.35654 + 0.504204i 7.94204 − 7.60294i 6.59135 − 6.12814i −4.61594 − 7.99504i
68.17 1.00088 + 0.577857i −2.23346 + 2.00292i −1.33216 − 2.30737i −3.81803 − 2.20434i −3.39282 + 0.714055i −4.15692 − 7.70206i 0.976660 − 8.94685i −2.54759 − 4.41256i
68.18 1.10869 + 0.640101i −2.99875 − 0.0866492i −1.18054 − 2.04476i 2.95232 + 1.70452i −3.26921 − 2.01557i 11.8984 − 8.14347i 8.98498 + 0.519678i 2.18213 + 3.77956i
68.19 1.61177 + 0.930557i −1.33950 − 2.68435i −0.268128 − 0.464411i −4.76252 − 2.74964i 0.338963 − 5.57304i −5.67067 − 8.44249i −5.41146 + 7.19139i −5.11740 − 8.86359i
68.20 1.64494 + 0.949706i 1.42058 − 2.64234i −0.196117 − 0.339685i 5.82785 + 3.36471i 4.84621 − 2.99736i −4.34657 − 8.34266i −4.96393 − 7.50729i 6.39097 + 11.0695i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.26
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.u.a yes 52
3.b odd 2 1 351.3.u.a 52
9.c even 3 1 351.3.k.a 52
9.d odd 6 1 117.3.k.a ✓ 52
13.c even 3 1 117.3.k.a ✓ 52
39.i odd 6 1 351.3.k.a 52
117.f even 3 1 351.3.u.a 52
117.u odd 6 1 inner 117.3.u.a yes 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.k.a ✓ 52 9.d odd 6 1
117.3.k.a ✓ 52 13.c even 3 1
117.3.u.a yes 52 1.a even 1 1 trivial
117.3.u.a yes 52 117.u odd 6 1 inner
351.3.k.a 52 9.c even 3 1
351.3.k.a 52 39.i odd 6 1
351.3.u.a 52 3.b odd 2 1
351.3.u.a 52 117.f even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).