Properties

Label 387.3.i.a
Level $387$
Weight $3$
Character orbit 387.i
Analytic conductor $10.545$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,3,Mod(251,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.251");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.i (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: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 128 q^{4} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 128 q^{4} - 6 q^{7} - 8 q^{10} + 34 q^{13} + 216 q^{16} - 32 q^{19} + 72 q^{22} + 142 q^{25} - 12 q^{28} + 74 q^{31} - 16 q^{34} - 2 q^{37} - 164 q^{40} + 40 q^{43} - 256 q^{46} - 56 q^{49} - 468 q^{52} + 336 q^{55} + 136 q^{58} - 204 q^{61} - 416 q^{64} - 366 q^{67} - 504 q^{70} - 182 q^{73} + 168 q^{76} + 196 q^{79} + 1144 q^{82} - 512 q^{85} + 40 q^{88} - 138 q^{91} + 1040 q^{94} + 880 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 3.82582i 0 −10.6369 −4.07908 + 2.35506i 0 −0.828985 + 1.43584i 25.3915i 0 9.01003 + 15.6058i
251.2 3.67873i 0 −9.53307 5.52313 3.18878i 0 4.16366 7.21167i 20.3547i 0 −11.7307 20.3181i
251.3 3.60468i 0 −8.99370 −6.10136 + 3.52262i 0 −1.11308 + 1.92790i 18.0007i 0 12.6979 + 21.9934i
251.4 3.08493i 0 −5.51676 0.670722 0.387242i 0 0.962002 1.66624i 4.67910i 0 −1.19461 2.06913i
251.5 2.92209i 0 −4.53862 1.25945 0.727146i 0 −4.31156 + 7.46785i 1.57390i 0 −2.12479 3.68024i
251.6 2.91777i 0 −4.51337 8.38570 4.84149i 0 −4.59727 + 7.96270i 1.49789i 0 −14.1263 24.4675i
251.7 2.74538i 0 −3.53712 −2.82066 + 1.62851i 0 0.221075 0.382913i 1.27079i 0 4.47088 + 7.74379i
251.8 2.33960i 0 −1.47375 −3.85397 + 2.22509i 0 1.93315 3.34832i 5.91043i 0 5.20583 + 9.01676i
251.9 2.00047i 0 −0.00186107 4.64054 2.67922i 0 6.65424 11.5255i 7.99814i 0 −5.35968 9.28324i
251.10 1.57119i 0 1.53137 4.22007 2.43646i 0 3.93613 6.81758i 8.69082i 0 −3.82813 6.63051i
251.11 1.38607i 0 2.07881 −1.93550 + 1.11746i 0 −5.77497 + 10.0025i 8.42566i 0 1.54888 + 2.68274i
251.12 1.15908i 0 2.65654 3.71486 2.14478i 0 −4.06824 + 7.04639i 7.71545i 0 −2.48596 4.30582i
251.13 1.07818i 0 2.83753 −7.10171 + 4.10018i 0 2.44369 4.23259i 7.37208i 0 4.42072 + 7.65691i
251.14 0.560654i 0 3.68567 −5.94504 + 3.43237i 0 −2.88161 + 4.99110i 4.30900i 0 1.92437 + 3.33311i
251.15 0.211632i 0 3.95521 3.50663 2.02455i 0 1.76176 3.05147i 1.68358i 0 −0.428460 0.742115i
251.16 0.211632i 0 3.95521 −3.50663 + 2.02455i 0 1.76176 3.05147i 1.68358i 0 −0.428460 0.742115i
251.17 0.560654i 0 3.68567 5.94504 3.43237i 0 −2.88161 + 4.99110i 4.30900i 0 1.92437 + 3.33311i
251.18 1.07818i 0 2.83753 7.10171 4.10018i 0 2.44369 4.23259i 7.37208i 0 4.42072 + 7.65691i
251.19 1.15908i 0 2.65654 −3.71486 + 2.14478i 0 −4.06824 + 7.04639i 7.71545i 0 −2.48596 4.30582i
251.20 1.38607i 0 2.07881 1.93550 1.11746i 0 −5.77497 + 10.0025i 8.42566i 0 1.54888 + 2.68274i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.c even 3 1 inner
129.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.i.a 60
3.b odd 2 1 inner 387.3.i.a 60
43.c even 3 1 inner 387.3.i.a 60
129.f odd 6 1 inner 387.3.i.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.3.i.a 60 1.a even 1 1 trivial
387.3.i.a 60 3.b odd 2 1 inner
387.3.i.a 60 43.c even 3 1 inner
387.3.i.a 60 129.f odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(387, [\chi])\).