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Properties

Label 187.3.f.a
Level $187$
Weight $3$
Character orbit 187.f
Analytic conductor $5.095$
Analytic rank $0$
Dimension $68$
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] = [187,3,Mod(21,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.21");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 187.f (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: \(5.09538094354\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 4 q^{3} + 120 q^{4} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 4 q^{3} + 120 q^{4} - 8 q^{5} + 14 q^{11} - 8 q^{12} - 28 q^{14} + 184 q^{16} - 156 q^{20} + 10 q^{22} - 132 q^{23} - 140 q^{27} + 72 q^{31} - 180 q^{33} - 108 q^{34} - 60 q^{37} + 64 q^{38} + 62 q^{44} + 192 q^{45} - 12 q^{47} - 136 q^{48} - 264 q^{55} - 240 q^{56} + 376 q^{58} + 104 q^{64} + 508 q^{67} - 224 q^{69} + 172 q^{71} - 196 q^{75} + 852 q^{78} - 1376 q^{80} + 580 q^{81} + 80 q^{82} - 88 q^{86} + 254 q^{88} - 20 q^{89} + 408 q^{91} - 1284 q^{92} - 196 q^{97} + 82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −3.78342 −1.63466 + 1.63466i 10.3143 −6.26403 + 6.26403i 6.18462 − 6.18462i −3.23856 + 3.23856i −23.8895 3.65575i 23.6994 − 23.6994i
21.2 −3.67619 −1.95585 + 1.95585i 9.51441 3.83314 − 3.83314i 7.19009 − 7.19009i 2.57998 − 2.57998i −20.2720 1.34930i −14.0914 + 14.0914i
21.3 −3.47025 3.60887 − 3.60887i 8.04262 −5.41199 + 5.41199i −12.5237 + 12.5237i 2.39178 − 2.39178i −14.0289 − 17.0478i 18.7810 − 18.7810i
21.4 −3.25912 2.21273 − 2.21273i 6.62186 2.73848 − 2.73848i −7.21155 + 7.21155i 7.64287 − 7.64287i −8.54495 − 0.792351i −8.92502 + 8.92502i
21.5 −3.24080 0.633931 − 0.633931i 6.50281 0.952655 − 0.952655i −2.05445 + 2.05445i −4.47079 + 4.47079i −8.11110 8.19626i −3.08737 + 3.08737i
21.6 −2.77149 −3.54255 + 3.54255i 3.68118 −0.535697 + 0.535697i 9.81815 − 9.81815i 3.80470 − 3.80470i 0.883617 − 16.0993i 1.48468 − 1.48468i
21.7 −2.67995 2.09069 − 2.09069i 3.18215 −0.371515 + 0.371515i −5.60295 + 5.60295i −3.99689 + 3.99689i 2.19181 0.258022i 0.995644 − 0.995644i
21.8 −2.37870 −2.29315 + 2.29315i 1.65823 5.59269 − 5.59269i 5.45472 − 5.45472i −9.56883 + 9.56883i 5.57038 − 1.51707i −13.3034 + 13.3034i
21.9 −2.19732 −0.931598 + 0.931598i 0.828204 −1.89499 + 1.89499i 2.04702 − 2.04702i 7.69260 − 7.69260i 6.96944 7.26425i 4.16390 − 4.16390i
21.10 −1.92183 −0.766471 + 0.766471i −0.306580 −4.69374 + 4.69374i 1.47303 − 1.47303i −2.28573 + 2.28573i 8.27650 7.82504i 9.02056 − 9.02056i
21.11 −1.68037 2.04149 − 2.04149i −1.17637 6.92650 − 6.92650i −3.43045 + 3.43045i 3.06001 − 3.06001i 8.69820 0.664655i −11.6391 + 11.6391i
21.12 −1.43651 3.99651 − 3.99651i −1.93645 1.17234 − 1.17234i −5.74101 + 5.74101i −2.58979 + 2.58979i 8.52775 − 22.9441i −1.68408 + 1.68408i
21.13 −1.02807 −2.84342 + 2.84342i −2.94308 4.08065 − 4.08065i 2.92323 − 2.92323i 3.11504 − 3.11504i 7.13795 − 7.17011i −4.19518 + 4.19518i
21.14 −0.933413 2.25557 − 2.25557i −3.12874 −6.18187 + 6.18187i −2.10537 + 2.10537i 5.28589 − 5.28589i 6.65406 − 1.17515i 5.77024 − 5.77024i
21.15 −0.868636 −3.38460 + 3.38460i −3.24547 −3.50596 + 3.50596i 2.93998 − 2.93998i −5.68431 + 5.68431i 6.29368 − 13.9110i 3.04540 − 3.04540i
21.16 −0.444250 −0.391412 + 0.391412i −3.80264 1.59265 − 1.59265i 0.173885 − 0.173885i 0.773308 − 0.773308i 3.46633 8.69359i −0.707535 + 0.707535i
21.17 −0.440021 1.90394 − 1.90394i −3.80638 −0.0293118 + 0.0293118i −0.837772 + 0.837772i −7.94954 + 7.94954i 3.43497 1.75006i 0.0128978 − 0.0128978i
21.18 0.440021 1.90394 − 1.90394i −3.80638 −0.0293118 + 0.0293118i 0.837772 − 0.837772i 7.94954 − 7.94954i −3.43497 1.75006i −0.0128978 + 0.0128978i
21.19 0.444250 −0.391412 + 0.391412i −3.80264 1.59265 − 1.59265i −0.173885 + 0.173885i −0.773308 + 0.773308i −3.46633 8.69359i 0.707535 − 0.707535i
21.20 0.868636 −3.38460 + 3.38460i −3.24547 −3.50596 + 3.50596i −2.93998 + 2.93998i 5.68431 − 5.68431i −6.29368 − 13.9110i −3.04540 + 3.04540i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
17.c even 4 1 inner
187.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.3.f.a ✓ 68
11.b odd 2 1 inner 187.3.f.a ✓ 68
17.c even 4 1 inner 187.3.f.a ✓ 68
187.f odd 4 1 inner 187.3.f.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.3.f.a ✓ 68 1.a even 1 1 trivial
187.3.f.a ✓ 68 11.b odd 2 1 inner
187.3.f.a ✓ 68 17.c even 4 1 inner
187.3.f.a ✓ 68 187.f odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(187, [\chi])\).