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Properties

Label 156.3.l.c
Level $156$
Weight $3$
Character orbit 156.l
Analytic conductor $4.251$
Analytic rank $0$
Dimension $96$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,3,Mod(47,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 156.l (of order \(4\), 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: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 36 q^{6} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 36 q^{6} - 64 q^{9} - 8 q^{13} + 80 q^{16} + 48 q^{18} + 8 q^{21} + 124 q^{24} - 8 q^{28} + 24 q^{33} + 64 q^{34} - 128 q^{37} - 136 q^{40} - 140 q^{42} - 160 q^{45} + 88 q^{46} - 108 q^{48} - 320 q^{52} - 216 q^{54} + 136 q^{57} + 280 q^{58} + 80 q^{60} - 464 q^{61} - 240 q^{66} - 120 q^{70} - 32 q^{72} + 288 q^{73} + 88 q^{76} - 296 q^{78} - 144 q^{81} + 204 q^{84} + 848 q^{85} - 392 q^{93} + 1384 q^{94} - 204 q^{96} + 336 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.99970 − 0.0345636i −2.79461 − 1.09094i 3.99761 + 0.138234i 0.206813 − 0.206813i 5.55068 + 2.27814i −5.52432 − 5.52432i −7.98925 − 0.414598i 6.61972 + 6.09749i −0.420713 + 0.406417i
47.2 −1.99928 − 0.0538061i −1.71704 + 2.46004i 3.99421 + 0.215147i 3.36591 − 3.36591i 3.56519 − 4.82591i −1.28244 − 1.28244i −7.97395 − 0.645050i −3.10358 − 8.44795i −6.91049 + 6.54828i
47.3 −1.98966 − 0.203083i 2.62747 − 1.44789i 3.91751 + 0.808132i 6.57239 − 6.57239i −5.52182 + 2.34723i 6.49081 + 6.49081i −7.63041 − 2.40349i 4.80721 − 7.60860i −14.4116 + 11.7421i
47.4 −1.98192 − 0.268345i 2.11848 − 2.12416i 3.85598 + 1.06368i −3.86746 + 3.86746i −4.76866 + 3.64142i −6.41212 − 6.41212i −7.35680 − 3.14285i −0.0240825 − 8.99997i 8.70279 − 6.62716i
47.5 −1.89206 + 0.648159i −0.777110 − 2.89760i 3.15978 − 2.45271i −0.949668 + 0.949668i 3.34845 + 4.97875i 7.87350 + 7.87350i −4.38874 + 6.68872i −7.79220 + 4.50351i 1.18129 − 2.41236i
47.6 −1.87876 + 0.685761i 2.74902 + 1.20121i 3.05946 − 2.57676i −2.23749 + 2.23749i −5.98848 − 0.371622i 1.91723 + 1.91723i −3.98095 + 6.93916i 6.11417 + 6.60431i 2.66932 − 5.73809i
47.7 −1.85319 − 0.752133i 1.00525 + 2.82656i 2.86859 + 2.78769i −2.55696 + 2.55696i 0.263035 − 5.99423i 4.54470 + 4.54470i −3.21932 − 7.32366i −6.97894 + 5.68282i 6.66170 − 2.81535i
47.8 −1.72189 − 1.01740i −1.05328 − 2.80902i 1.92978 + 3.50370i −1.91537 + 1.91537i −1.04429 + 5.90842i 1.82665 + 1.82665i 0.241817 − 7.99634i −6.78121 + 5.91736i 5.24676 − 1.34935i
47.9 −1.72150 + 1.01806i −1.28511 + 2.71081i 1.92709 − 3.50518i −5.44277 + 5.44277i −0.547465 − 5.97497i −2.75895 − 2.75895i 0.251018 + 7.99606i −5.69698 − 6.96738i 3.82862 − 14.9108i
47.10 −1.63658 − 1.14961i 2.64742 + 1.41109i 1.35681 + 3.76285i 3.10286 − 3.10286i −2.71053 − 5.35285i −7.50740 − 7.50740i 2.10527 − 7.71802i 5.01767 + 7.47148i −8.64517 + 1.51102i
47.11 −1.62136 + 1.17098i −0.352823 − 2.97918i 1.25761 − 3.79716i 5.05107 − 5.05107i 4.06062 + 4.41717i −7.03089 − 7.03089i 2.40736 + 7.62919i −8.75103 + 2.10225i −2.27489 + 14.1043i
47.12 −1.52401 + 1.29514i −2.96389 + 0.464078i 0.645205 − 3.94762i 1.61076 − 1.61076i 3.91594 − 4.54592i 3.94036 + 3.94036i 4.12944 + 6.85184i 8.56926 − 2.75095i −0.368647 + 4.54099i
47.13 −1.29514 + 1.52401i 2.96389 − 0.464078i −0.645205 − 3.94762i 1.61076 − 1.61076i −3.13140 + 5.11804i −3.94036 − 3.94036i 6.85184 + 4.12944i 8.56926 − 2.75095i 0.368647 + 4.54099i
47.14 −1.17098 + 1.62136i 0.352823 + 2.97918i −1.25761 − 3.79716i 5.05107 − 5.05107i −5.24347 − 2.91651i 7.03089 + 7.03089i 7.62919 + 2.40736i −8.75103 + 2.10225i 2.27489 + 14.1043i
47.15 −1.14961 − 1.63658i 2.64742 − 1.41109i −1.35681 + 3.76285i −3.10286 + 3.10286i −5.35285 − 2.71053i 7.50740 + 7.50740i 7.71802 − 2.10527i 5.01767 − 7.47148i 8.64517 + 1.51102i
47.16 −1.01806 + 1.72150i 1.28511 − 2.71081i −1.92709 − 3.50518i −5.44277 + 5.44277i 3.35832 + 4.97209i 2.75895 + 2.75895i 7.99606 + 0.251018i −5.69698 − 6.96738i −3.82862 − 14.9108i
47.17 −1.01740 − 1.72189i −1.05328 + 2.80902i −1.92978 + 3.50370i 1.91537 − 1.91537i 5.90842 − 1.04429i −1.82665 − 1.82665i 7.99634 − 0.241817i −6.78121 − 5.91736i −5.24676 − 1.34935i
47.18 −0.752133 − 1.85319i 1.00525 − 2.82656i −2.86859 + 2.78769i 2.55696 − 2.55696i −5.99423 + 0.263035i −4.54470 − 4.54470i 7.32366 + 3.21932i −6.97894 − 5.68282i −6.66170 − 2.81535i
47.19 −0.685761 + 1.87876i −2.74902 − 1.20121i −3.05946 − 2.57676i −2.23749 + 2.23749i 4.14196 − 4.34099i −1.91723 − 1.91723i 6.93916 − 3.98095i 6.11417 + 6.60431i −2.66932 − 5.73809i
47.20 −0.648159 + 1.89206i 0.777110 + 2.89760i −3.15978 − 2.45271i −0.949668 + 0.949668i −5.98613 − 0.407769i −7.87350 − 7.87350i 6.68872 − 4.38874i −7.79220 + 4.50351i −1.18129 − 2.41236i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner
52.f even 4 1 inner
156.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.l.c ✓ 96
3.b odd 2 1 inner 156.3.l.c ✓ 96
4.b odd 2 1 inner 156.3.l.c ✓ 96
12.b even 2 1 inner 156.3.l.c ✓ 96
13.d odd 4 1 inner 156.3.l.c ✓ 96
39.f even 4 1 inner 156.3.l.c ✓ 96
52.f even 4 1 inner 156.3.l.c ✓ 96
156.l odd 4 1 inner 156.3.l.c ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.l.c ✓ 96 1.a even 1 1 trivial
156.3.l.c ✓ 96 3.b odd 2 1 inner
156.3.l.c ✓ 96 4.b odd 2 1 inner
156.3.l.c ✓ 96 12.b even 2 1 inner
156.3.l.c ✓ 96 13.d odd 4 1 inner
156.3.l.c ✓ 96 39.f even 4 1 inner
156.3.l.c ✓ 96 52.f even 4 1 inner
156.3.l.c ✓ 96 156.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(156, [\chi])\):

\( T_{5}^{48} + 15712 T_{5}^{44} + 85407078 T_{5}^{40} + 208188820460 T_{5}^{36} + 244444625842321 T_{5}^{32} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
\( T_{7}^{48} + 58452 T_{7}^{44} + 1404716438 T_{7}^{40} + 17986204748884 T_{7}^{36} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display