Properties

Label 228.3.o.c
Level $228$
Weight $3$
Character orbit 228.o
Analytic conductor $6.213$
Analytic rank $0$
Dimension $24$
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] = [228,3,Mod(125,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.125");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.o (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: \(6.21255002741\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{3} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{3} + 8 q^{7} + 16 q^{9} + 10 q^{13} - 11 q^{15} - 6 q^{19} - 24 q^{21} + 138 q^{25} + 196 q^{27} - 80 q^{31} + 53 q^{33} - 16 q^{37} + 54 q^{39} + 142 q^{43} - 154 q^{45} - 288 q^{49} - 23 q^{51} - 120 q^{55} - 213 q^{57} - 30 q^{61} - 20 q^{63} - 140 q^{67} - 142 q^{69} + 224 q^{73} + 162 q^{75} + 122 q^{79} + 40 q^{81} + 18 q^{85} - 394 q^{87} - 292 q^{91} + 130 q^{93} - 132 q^{97} - 197 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 0 −2.99994 0.0182875i 0 1.08094 + 0.624082i 0 −4.29950 0 8.99933 + 0.109723i 0
125.2 0 −2.60151 1.49404i 0 −0.973208 0.561882i 0 6.36523 0 4.53572 + 7.77350i 0
125.3 0 −2.35298 + 1.86104i 0 2.07590 + 1.19852i 0 −6.50893 0 2.07305 8.75799i 0
125.4 0 −2.27725 + 1.95298i 0 −7.75465 4.47715i 0 10.5834 0 1.37176 8.89485i 0
125.5 0 −1.73353 2.44845i 0 8.34937 + 4.82051i 0 −1.90607 0 −2.98977 + 8.48889i 0
125.6 0 −1.11376 2.78560i 0 −5.28947 3.05387i 0 −2.23411 0 −6.51909 + 6.20495i 0
125.7 0 −0.552701 + 2.94865i 0 7.75465 + 4.47715i 0 10.5834 0 −8.38904 3.25944i 0
125.8 0 −0.435218 + 2.96826i 0 −2.07590 1.19852i 0 −6.50893 0 −8.62117 2.58368i 0
125.9 0 1.51581 + 2.58888i 0 −1.08094 0.624082i 0 −4.29950 0 −4.40464 + 7.84851i 0
125.10 0 2.59463 + 1.50596i 0 0.973208 + 0.561882i 0 6.36523 0 4.46419 + 7.81480i 0
125.11 0 2.96928 0.428257i 0 5.28947 + 3.05387i 0 −2.23411 0 8.63319 2.54322i 0
125.12 0 2.98718 + 0.277055i 0 −8.34937 4.82051i 0 −1.90607 0 8.84648 + 1.65522i 0
197.1 0 −2.99994 + 0.0182875i 0 1.08094 0.624082i 0 −4.29950 0 8.99933 0.109723i 0
197.2 0 −2.60151 + 1.49404i 0 −0.973208 + 0.561882i 0 6.36523 0 4.53572 7.77350i 0
197.3 0 −2.35298 1.86104i 0 2.07590 1.19852i 0 −6.50893 0 2.07305 + 8.75799i 0
197.4 0 −2.27725 1.95298i 0 −7.75465 + 4.47715i 0 10.5834 0 1.37176 + 8.89485i 0
197.5 0 −1.73353 + 2.44845i 0 8.34937 4.82051i 0 −1.90607 0 −2.98977 8.48889i 0
197.6 0 −1.11376 + 2.78560i 0 −5.28947 + 3.05387i 0 −2.23411 0 −6.51909 6.20495i 0
197.7 0 −0.552701 2.94865i 0 7.75465 4.47715i 0 10.5834 0 −8.38904 + 3.25944i 0
197.8 0 −0.435218 2.96826i 0 −2.07590 + 1.19852i 0 −6.50893 0 −8.62117 + 2.58368i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.3.o.c 24
3.b odd 2 1 inner 228.3.o.c 24
19.c even 3 1 inner 228.3.o.c 24
57.h odd 6 1 inner 228.3.o.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.o.c 24 1.a even 1 1 trivial
228.3.o.c 24 3.b odd 2 1 inner
228.3.o.c 24 19.c even 3 1 inner
228.3.o.c 24 57.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\):

\( T_{5}^{24} - 219 T_{5}^{22} + 32229 T_{5}^{20} - 2643260 T_{5}^{18} + 157034700 T_{5}^{16} - 5159171592 T_{5}^{14} + 118202291248 T_{5}^{12} - 894201641520 T_{5}^{10} + \cdots + 9877191840000 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} - 109T_{7}^{4} - 136T_{7}^{3} + 2562T_{7}^{2} + 8886T_{7} + 8028 \) Copy content Toggle raw display