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Properties

Label 228.3.r.b
Level $228$
Weight $3$
Character orbit 228.r
Analytic conductor $6.213$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [228,3,Mod(13,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.r (of order \(18\), degree \(6\), 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: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{7} + 15 q^{11} - 9 q^{13} - 18 q^{15} + 18 q^{17} + 33 q^{19} + 18 q^{21} - 21 q^{23} - 18 q^{25} + 108 q^{27} + 30 q^{29} + 216 q^{31} - 63 q^{33} + 30 q^{35} + 18 q^{39} - 54 q^{41} - 189 q^{43} - 9 q^{45} - 321 q^{47} - 225 q^{49} + 63 q^{51} + 138 q^{53} + 309 q^{55} - 45 q^{57} - 30 q^{59} - 105 q^{61} - 27 q^{63} + 99 q^{65} + 207 q^{67} + 108 q^{69} + 144 q^{71} - 72 q^{73} - 36 q^{77} - 282 q^{79} - 531 q^{83} - 909 q^{85} - 135 q^{87} - 411 q^{89} + 444 q^{91} + 126 q^{93} - 24 q^{95} + 405 q^{97} - 117 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −0.592396 + 1.62760i 0 −1.20343 − 6.82500i 0 −4.14835 + 7.18516i 0 −2.29813 − 1.92836i 0
13.2 0 −0.592396 + 1.62760i 0 −0.523375 − 2.96821i 0 2.50080 − 4.33150i 0 −2.29813 − 1.92836i 0
13.3 0 −0.592396 + 1.62760i 0 0.726964 + 4.12282i 0 −4.76521 + 8.25358i 0 −2.29813 − 1.92836i 0
13.4 0 −0.592396 + 1.62760i 0 1.41859 + 8.04523i 0 6.02610 − 10.4375i 0 −2.29813 − 1.92836i 0
97.1 0 −1.11334 + 1.32683i 0 −6.97742 + 2.53957i 0 −0.260735 − 0.451607i 0 −0.520945 − 2.95442i 0
97.2 0 −1.11334 + 1.32683i 0 −1.60704 + 0.584914i 0 2.14783 + 3.72014i 0 −0.520945 − 2.95442i 0
97.3 0 −1.11334 + 1.32683i 0 2.90659 − 1.05791i 0 1.09476 + 1.89618i 0 −0.520945 − 2.95442i 0
97.4 0 −1.11334 + 1.32683i 0 7.73091 − 2.81382i 0 −6.18759 − 10.7172i 0 −0.520945 − 2.95442i 0
109.1 0 1.70574 + 0.300767i 0 −4.52547 − 3.79732i 0 −5.02788 − 8.70855i 0 2.81908 + 1.02606i 0
109.2 0 1.70574 + 0.300767i 0 −3.64665 − 3.05991i 0 6.09878 + 10.5634i 0 2.81908 + 1.02606i 0
109.3 0 1.70574 + 0.300767i 0 −0.168919 − 0.141740i 0 −3.45586 − 5.98572i 0 2.81908 + 1.02606i 0
109.4 0 1.70574 + 0.300767i 0 5.86927 + 4.92490i 0 1.47736 + 2.55885i 0 2.81908 + 1.02606i 0
181.1 0 −1.11334 − 1.32683i 0 −6.97742 − 2.53957i 0 −0.260735 + 0.451607i 0 −0.520945 + 2.95442i 0
181.2 0 −1.11334 − 1.32683i 0 −1.60704 − 0.584914i 0 2.14783 − 3.72014i 0 −0.520945 + 2.95442i 0
181.3 0 −1.11334 − 1.32683i 0 2.90659 + 1.05791i 0 1.09476 − 1.89618i 0 −0.520945 + 2.95442i 0
181.4 0 −1.11334 − 1.32683i 0 7.73091 + 2.81382i 0 −6.18759 + 10.7172i 0 −0.520945 + 2.95442i 0
193.1 0 −0.592396 − 1.62760i 0 −1.20343 + 6.82500i 0 −4.14835 − 7.18516i 0 −2.29813 + 1.92836i 0
193.2 0 −0.592396 − 1.62760i 0 −0.523375 + 2.96821i 0 2.50080 + 4.33150i 0 −2.29813 + 1.92836i 0
193.3 0 −0.592396 − 1.62760i 0 0.726964 − 4.12282i 0 −4.76521 − 8.25358i 0 −2.29813 + 1.92836i 0
193.4 0 −0.592396 − 1.62760i 0 1.41859 − 8.04523i 0 6.02610 + 10.4375i 0 −2.29813 + 1.92836i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.3.r.b ✓ 24
19.f odd 18 1 inner 228.3.r.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.r.b ✓ 24 1.a even 1 1 trivial
228.3.r.b ✓ 24 19.f odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 9 T_{5}^{22} + 74 T_{5}^{21} - 936 T_{5}^{20} + 1467 T_{5}^{19} + 185601 T_{5}^{18} + \cdots + 120293338780224 \) acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display