↑

Properties

Label 117.2.bc.a
Level $117$
Weight $2$
Character orbit 117.bc
Analytic conductor $0.934$
Analytic rank $0$
Dimension $48$
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(20,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.bc (of order \(12\), degree \(4\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} - 2 q^{6} + 2 q^{7} - 30 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} - 2 q^{3} - 6 q^{4} - 6 q^{5} - 2 q^{6} + 2 q^{7} - 30 q^{8} - 2 q^{9} - 12 q^{10} + 6 q^{11} - 18 q^{12} - 2 q^{13} - 12 q^{14} + 4 q^{15} + 14 q^{16} - 2 q^{18} - 4 q^{19} - 6 q^{20} + 22 q^{21} + 2 q^{22} - 12 q^{23} - 18 q^{24} + 48 q^{26} - 32 q^{27} - 6 q^{29} + 66 q^{30} + 6 q^{31} + 30 q^{32} - 56 q^{33} - 6 q^{34} - 6 q^{35} - 6 q^{36} - 6 q^{37} - 36 q^{38} - 32 q^{39} - 12 q^{40} + 18 q^{41} + 80 q^{42} - 12 q^{44} + 34 q^{45} - 12 q^{46} + 30 q^{47} + 22 q^{48} - 12 q^{50} - 16 q^{52} - 56 q^{54} - 4 q^{55} - 12 q^{56} - 2 q^{57} - 28 q^{58} + 30 q^{59} - 58 q^{60} - 4 q^{61} - 18 q^{62} - 2 q^{63} + 30 q^{65} + 32 q^{66} - 16 q^{67} - 48 q^{69} - 46 q^{70} + 48 q^{71} + 126 q^{72} - 22 q^{73} + 24 q^{75} - 18 q^{76} - 72 q^{77} + 94 q^{78} + 8 q^{79} + 54 q^{80} - 14 q^{81} - 12 q^{82} + 72 q^{83} - 110 q^{84} + 78 q^{85} + 102 q^{86} + 14 q^{87} - 6 q^{88} + 114 q^{90} - 16 q^{91} + 120 q^{92} - 44 q^{93} - 52 q^{94} - 6 q^{95} + 16 q^{96} + 48 q^{97} - 36 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −2.60753 − 0.698685i −1.12895 − 1.31357i 4.57899 + 2.64368i 1.93643 + 0.518864i 2.02600 + 4.21396i 1.26724 − 1.26724i −6.27505 − 6.27505i −0.450943 + 2.96591i −4.68676 − 2.70590i
20.2 −2.30434 − 0.617447i 0.895864 + 1.48237i 3.19671 + 1.84562i −0.605981 − 0.162372i −1.14909 − 3.96904i −1.66339 + 1.66339i −2.85295 − 2.85295i −1.39486 + 2.65601i 1.29613 + 0.748322i
20.3 −1.74626 − 0.467910i 0.825450 − 1.52271i 1.09844 + 0.634187i −3.66400 − 0.981766i −2.15394 + 2.27281i −0.671277 + 0.671277i 0.935276 + 0.935276i −1.63726 − 2.51383i 5.93893 + 3.42884i
20.4 −1.55681 − 0.417145i −1.30216 + 1.14210i 0.517589 + 0.298830i −0.690114 − 0.184916i 2.50363 − 1.23483i 3.10547 − 3.10547i 1.59819 + 1.59819i 0.391235 − 2.97438i 0.997239 + 0.575756i
20.5 −1.03000 − 0.275986i −1.73204 + 0.00536576i −0.747329 − 0.431471i 2.22216 + 0.595426i 1.78548 + 0.472494i −3.66177 + 3.66177i 2.15868 + 2.15868i 2.99994 − 0.0185875i −2.12448 − 1.22657i
20.6 −0.724963 − 0.194253i 1.63234 + 0.579188i −1.24421 − 0.718347i 0.862152 + 0.231013i −1.07088 − 0.736978i 2.34485 − 2.34485i 1.82389 + 1.82389i 2.32908 + 1.89087i −0.580153 − 0.334952i
20.7 0.0826047 + 0.0221339i 0.405260 − 1.68397i −1.72572 − 0.996343i 1.73943 + 0.466078i 0.0707492 − 0.130134i 0.362366 − 0.362366i −0.241441 − 0.241441i −2.67153 − 1.36489i 0.133369 + 0.0770004i
20.8 0.425586 + 0.114035i −1.46672 − 0.921268i −1.56393 − 0.902936i −2.94562 − 0.789277i −0.519158 − 0.559337i 0.205334 − 0.205334i −1.18572 − 1.18572i 1.30253 + 2.70248i −1.16361 − 0.671810i
20.9 0.863432 + 0.231356i 0.168175 + 1.72387i −1.04006 − 0.600479i 2.81220 + 0.753527i −0.253619 + 1.52735i −0.653306 + 0.653306i −2.02325 − 2.02325i −2.94343 + 0.579822i 2.25381 + 1.30124i
20.10 1.30411 + 0.349436i 1.57241 − 0.726303i −0.153448 − 0.0885935i −0.236109 − 0.0632653i 2.30440 − 0.397723i −2.19861 + 2.19861i −2.07851 − 2.07851i 1.94497 − 2.28410i −0.285806 − 0.165010i
20.11 1.86810 + 0.500557i 1.15921 + 1.28694i 1.50720 + 0.870184i −4.03603 − 1.08145i 1.52135 + 2.98439i 1.89104 − 1.89104i −0.355058 − 0.355058i −0.312441 + 2.98369i −6.99840 − 4.04053i
20.12 2.19401 + 0.587883i −1.52885 + 0.814015i 2.73602 + 1.57964i 0.239470 + 0.0641657i −3.83286 + 0.887173i −0.693990 + 0.693990i 1.86196 + 1.86196i 1.67476 − 2.48901i 0.487677 + 0.281560i
41.1 −2.60753 + 0.698685i −1.12895 + 1.31357i 4.57899 − 2.64368i 1.93643 − 0.518864i 2.02600 − 4.21396i 1.26724 + 1.26724i −6.27505 + 6.27505i −0.450943 − 2.96591i −4.68676 + 2.70590i
41.2 −2.30434 + 0.617447i 0.895864 − 1.48237i 3.19671 − 1.84562i −0.605981 + 0.162372i −1.14909 + 3.96904i −1.66339 − 1.66339i −2.85295 + 2.85295i −1.39486 − 2.65601i 1.29613 − 0.748322i
41.3 −1.74626 + 0.467910i 0.825450 + 1.52271i 1.09844 − 0.634187i −3.66400 + 0.981766i −2.15394 − 2.27281i −0.671277 − 0.671277i 0.935276 − 0.935276i −1.63726 + 2.51383i 5.93893 − 3.42884i
41.4 −1.55681 + 0.417145i −1.30216 − 1.14210i 0.517589 − 0.298830i −0.690114 + 0.184916i 2.50363 + 1.23483i 3.10547 + 3.10547i 1.59819 − 1.59819i 0.391235 + 2.97438i 0.997239 − 0.575756i
41.5 −1.03000 + 0.275986i −1.73204 − 0.00536576i −0.747329 + 0.431471i 2.22216 − 0.595426i 1.78548 − 0.472494i −3.66177 − 3.66177i 2.15868 − 2.15868i 2.99994 + 0.0185875i −2.12448 + 1.22657i
41.6 −0.724963 + 0.194253i 1.63234 − 0.579188i −1.24421 + 0.718347i 0.862152 − 0.231013i −1.07088 + 0.736978i 2.34485 + 2.34485i 1.82389 − 1.82389i 2.32908 − 1.89087i −0.580153 + 0.334952i
41.7 0.0826047 − 0.0221339i 0.405260 + 1.68397i −1.72572 + 0.996343i 1.73943 − 0.466078i 0.0707492 + 0.130134i 0.362366 + 0.362366i −0.241441 + 0.241441i −2.67153 + 1.36489i 0.133369 − 0.0770004i
41.8 0.425586 − 0.114035i −1.46672 + 0.921268i −1.56393 + 0.902936i −2.94562 + 0.789277i −0.519158 + 0.559337i 0.205334 + 0.205334i −1.18572 + 1.18572i 1.30253 − 2.70248i −1.16361 + 0.671810i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.bc.a yes 48
3.b odd 2 1 351.2.bf.a 48
9.c even 3 1 351.2.ba.a 48
9.d odd 6 1 117.2.x.a ✓ 48
13.f odd 12 1 117.2.x.a ✓ 48
39.k even 12 1 351.2.ba.a 48
117.bb odd 12 1 351.2.bf.a 48
117.bc even 12 1 inner 117.2.bc.a yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.x.a ✓ 48 9.d odd 6 1
117.2.x.a ✓ 48 13.f odd 12 1
117.2.bc.a yes 48 1.a even 1 1 trivial
117.2.bc.a yes 48 117.bc even 12 1 inner
351.2.ba.a 48 9.c even 3 1
351.2.ba.a 48 39.k even 12 1
351.2.bf.a 48 3.b odd 2 1
351.2.bf.a 48 117.bb odd 12 1

Hecke kernels

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