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Properties

Label 187.2.m.a
Level $187$
Weight $2$
Character orbit 187.m
Analytic conductor $1.493$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [187,2,Mod(10,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 187.m (of order \(16\), degree \(8\), 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: \(1.49320251780\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9} - 64 q^{12} + 16 q^{14} - 16 q^{15} - 16 q^{20} + 8 q^{22} - 32 q^{23} + 16 q^{26} + 32 q^{27} - 80 q^{31} - 16 q^{34} + 16 q^{36} + 16 q^{37} + 80 q^{38} - 112 q^{42} - 72 q^{44} - 16 q^{45} - 16 q^{47} + 32 q^{48} + 48 q^{49} - 16 q^{53} - 8 q^{55} - 16 q^{56} - 128 q^{58} + 48 q^{59} + 112 q^{60} + 48 q^{64} + 72 q^{66} - 192 q^{69} + 128 q^{70} - 16 q^{71} + 48 q^{75} + 8 q^{77} - 16 q^{78} + 112 q^{80} - 16 q^{82} + 32 q^{86} - 168 q^{88} - 16 q^{89} + 144 q^{91} - 176 q^{92} + 48 q^{93} + 48 q^{97} - 112 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.31737 − 0.959888i 0.0998402 + 0.0667111i 3.03462 + 3.03462i −1.55498 + 0.309305i −0.167332 − 0.250430i −0.0889145 + 0.447003i −2.19968 − 5.31050i −1.14253 − 2.75832i 3.90037 + 0.775832i
10.2 −2.25162 − 0.932652i −2.66909 − 1.78343i 2.78574 + 2.78574i 2.54233 − 0.505701i 4.34645 + 6.50493i −0.607350 + 3.05336i −1.80901 − 4.36732i 2.79536 + 6.74860i −6.19601 − 1.23246i
10.3 −1.58527 − 0.656642i 2.26689 + 1.51468i 0.667703 + 0.667703i −3.74296 + 0.744522i −2.59903 − 3.88972i −0.334093 + 1.67960i 0.693234 + 1.67362i 1.69645 + 4.09559i 6.42251 + 1.27752i
10.4 −1.50318 − 0.622639i 1.02275 + 0.683380i 0.457668 + 0.457668i 0.908813 − 0.180774i −1.11188 − 1.66405i 0.274197 − 1.37848i 0.842281 + 2.03345i −0.569040 − 1.37378i −1.47867 − 0.294126i
10.5 −1.30126 − 0.539000i −2.38581 − 1.59415i −0.0114548 − 0.0114548i −3.07041 + 0.610742i 2.24531 + 3.36035i 0.785949 − 3.95123i 1.08673 + 2.62360i 2.00273 + 4.83502i 4.32459 + 0.860214i
10.6 −0.896657 − 0.371407i −1.08522 − 0.725122i −0.748164 − 0.748164i 2.89314 − 0.575481i 0.703755 + 1.05324i 0.0961645 − 0.483452i 1.13579 + 2.74203i −0.496147 − 1.19780i −2.80789 − 0.558524i
10.7 −0.821699 − 0.340359i −1.10085 − 0.735567i −0.854868 − 0.854868i −1.20971 + 0.240627i 0.654214 + 0.979100i −0.644166 + 3.23844i 1.09220 + 2.63681i −0.477230 − 1.15214i 1.07592 + 0.214014i
10.8 −0.198710 − 0.0823085i 2.14439 + 1.43283i −1.38150 − 1.38150i 1.69258 − 0.336675i −0.308177 − 0.461220i 0.860605 − 4.32655i 0.325427 + 0.785649i 1.39733 + 3.37345i −0.364045 − 0.0724130i
10.9 0.198710 + 0.0823085i 2.14439 + 1.43283i −1.38150 − 1.38150i 1.69258 − 0.336675i 0.308177 + 0.461220i −0.860605 + 4.32655i −0.325427 − 0.785649i 1.39733 + 3.37345i 0.364045 + 0.0724130i
10.10 0.821699 + 0.340359i −1.10085 − 0.735567i −0.854868 − 0.854868i −1.20971 + 0.240627i −0.654214 − 0.979100i 0.644166 − 3.23844i −1.09220 − 2.63681i −0.477230 − 1.15214i −1.07592 − 0.214014i
10.11 0.896657 + 0.371407i −1.08522 − 0.725122i −0.748164 − 0.748164i 2.89314 − 0.575481i −0.703755 − 1.05324i −0.0961645 + 0.483452i −1.13579 − 2.74203i −0.496147 − 1.19780i 2.80789 + 0.558524i
10.12 1.30126 + 0.539000i −2.38581 − 1.59415i −0.0114548 − 0.0114548i −3.07041 + 0.610742i −2.24531 − 3.36035i −0.785949 + 3.95123i −1.08673 − 2.62360i 2.00273 + 4.83502i −4.32459 − 0.860214i
10.13 1.50318 + 0.622639i 1.02275 + 0.683380i 0.457668 + 0.457668i 0.908813 − 0.180774i 1.11188 + 1.66405i −0.274197 + 1.37848i −0.842281 − 2.03345i −0.569040 − 1.37378i 1.47867 + 0.294126i
10.14 1.58527 + 0.656642i 2.26689 + 1.51468i 0.667703 + 0.667703i −3.74296 + 0.744522i 2.59903 + 3.88972i 0.334093 − 1.67960i −0.693234 − 1.67362i 1.69645 + 4.09559i −6.42251 − 1.27752i
10.15 2.25162 + 0.932652i −2.66909 − 1.78343i 2.78574 + 2.78574i 2.54233 − 0.505701i −4.34645 − 6.50493i 0.607350 − 3.05336i 1.80901 + 4.36732i 2.79536 + 6.74860i 6.19601 + 1.23246i
10.16 2.31737 + 0.959888i 0.0998402 + 0.0667111i 3.03462 + 3.03462i −1.55498 + 0.309305i 0.167332 + 0.250430i 0.0889145 − 0.447003i 2.19968 + 5.31050i −1.14253 − 2.75832i −3.90037 − 0.775832i
54.1 −1.05312 − 2.54247i 1.00237 + 0.199384i −3.94085 + 3.94085i −0.856163 − 1.28134i −0.548692 − 2.75846i −2.87065 − 1.91811i 9.08474 + 3.76302i −1.80665 − 0.748339i −2.35611 + 3.52617i
54.2 −0.926579 − 2.23696i −1.59986 − 0.318232i −2.73122 + 2.73122i 1.86996 + 2.79859i 0.770525 + 3.87369i 2.28141 + 1.52439i 4.16641 + 1.72579i −0.313354 − 0.129795i 4.52767 − 6.77614i
54.3 −0.791873 − 1.91175i 2.13852 + 0.425377i −1.61351 + 1.61351i 0.428580 + 0.641415i −0.880218 − 4.42515i 3.78229 + 2.52725i 0.538832 + 0.223192i 1.62067 + 0.671303i 0.886844 − 1.32726i
54.4 −0.657401 − 1.58711i −0.508365 − 0.101120i −0.672515 + 0.672515i 0.528726 + 0.791294i 0.173711 + 0.873306i −3.14310 − 2.10015i −1.66475 − 0.689561i −2.52343 − 1.04524i 0.908282 − 1.35934i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
17.e odd 16 1 inner
187.m even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.2.m.a ✓ 128
11.b odd 2 1 inner 187.2.m.a ✓ 128
17.e odd 16 1 inner 187.2.m.a ✓ 128
187.m even 16 1 inner 187.2.m.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.2.m.a ✓ 128 1.a even 1 1 trivial
187.2.m.a ✓ 128 11.b odd 2 1 inner
187.2.m.a ✓ 128 17.e odd 16 1 inner
187.2.m.a ✓ 128 187.m even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(187, [\chi])\).