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Properties

Label 228.3.x.a
Level $228$
Weight $3$
Character orbit 228.x
Analytic conductor $6.213$
Analytic rank $0$
Dimension $120$
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(43,228)]
 
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([9, 0, 16]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 228.x (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: \(120\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 9 q^{2} - 3 q^{4} - 9 q^{6} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 9 q^{2} - 3 q^{4} - 9 q^{6} + 27 q^{8} - 12 q^{10} + 12 q^{13} + 45 q^{14} + 21 q^{16} - 108 q^{19} - 30 q^{20} - 36 q^{21} + 108 q^{22} - 540 q^{27} + 162 q^{28} - 72 q^{31} + 201 q^{32} + 273 q^{34} + 9 q^{36} - 84 q^{38} - 189 q^{40} - 24 q^{41} - 135 q^{42} + 324 q^{43} - 21 q^{44} - 351 q^{46} - 432 q^{47} + 36 q^{48} + 420 q^{49} - 117 q^{50} - 195 q^{52} + 24 q^{53} + 27 q^{54} + 36 q^{58} + 99 q^{60} + 96 q^{61} - 135 q^{62} + 21 q^{64} + 120 q^{65} - 216 q^{66} + 252 q^{67} + 141 q^{68} + 144 q^{69} - 180 q^{70} + 432 q^{71} - 54 q^{72} - 180 q^{73} + 393 q^{74} - 90 q^{76} - 144 q^{77} + 207 q^{78} - 144 q^{80} - 891 q^{82} + 297 q^{84} - 456 q^{85} + 1017 q^{86} + 156 q^{88} - 360 q^{89} + 171 q^{90} + 1305 q^{92} - 216 q^{93} - 402 q^{94} + 144 q^{95} + 72 q^{97} + 339 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.99985 − 0.0245464i −1.70574 − 0.300767i 3.99879 + 0.0981783i −3.03716 − 2.54848i 3.40383 + 0.643359i 2.94345 − 1.69940i −7.99458 − 0.294498i 2.81908 + 1.02606i 6.01131 + 5.17113i
43.2 −1.98646 + 0.232340i −1.70574 − 0.300767i 3.89204 − 0.923068i 6.79139 + 5.69866i 3.45826 + 0.201151i 4.98039 − 2.87543i −7.51690 + 2.73791i 2.81908 + 1.02606i −14.8149 − 9.74223i
43.3 −1.85401 − 0.750102i −1.70574 − 0.300767i 2.87469 + 2.78139i −0.466896 − 0.391772i 2.93684 + 1.83710i −0.729386 + 0.421111i −3.24338 − 7.31303i 2.81908 + 1.02606i 0.571760 + 1.07657i
43.4 −1.71973 + 1.02104i −1.70574 − 0.300767i 1.91495 − 3.51183i −1.38001 − 1.15796i 3.24051 − 1.22439i −9.01783 + 5.20645i 0.292513 + 7.99465i 2.81908 + 1.02606i 3.55556 + 0.582343i
43.5 −1.51461 + 1.30612i −1.70574 − 0.300767i 0.588117 − 3.95653i −6.59608 − 5.53476i 2.97637 − 1.77235i 6.43330 − 3.71427i 4.27692 + 6.76077i 2.81908 + 1.02606i 17.2196 − 0.232210i
43.6 −1.26630 − 1.54806i −1.70574 − 0.300767i −0.792964 + 3.92061i 5.84781 + 4.90689i 1.69437 + 3.02144i −8.68081 + 5.01187i 7.07347 − 3.73712i 2.81908 + 1.02606i 0.191069 − 15.2663i
43.7 −1.06319 − 1.69400i −1.70574 − 0.300767i −1.73926 + 3.60208i −7.00754 − 5.88002i 1.30402 + 3.20929i −6.50879 + 3.75785i 7.95108 − 0.883392i 2.81908 + 1.02606i −2.51041 + 18.1223i
43.8 −0.838585 + 1.81570i −1.70574 − 0.300767i −2.59355 − 3.04524i 4.46803 + 3.74913i 1.97651 − 2.84489i −5.06785 + 2.92592i 7.70417 − 2.15542i 2.81908 + 1.02606i −10.5541 + 4.96866i
43.9 −0.811142 − 1.82813i −1.70574 − 0.300767i −2.68410 + 2.96574i −1.23114 − 1.03305i 0.833753 + 3.36227i 11.5569 − 6.67236i 7.59893 + 2.50124i 2.81908 + 1.02606i −0.889916 + 3.08863i
43.10 −0.106253 + 1.99718i −1.70574 − 0.300767i −3.97742 − 0.424413i −2.84885 − 2.39047i 0.781926 − 3.37470i 3.69500 − 2.13331i 1.27024 − 7.89851i 2.81908 + 1.02606i 5.07689 − 5.43566i
43.11 −0.0474309 − 1.99944i −1.70574 − 0.300767i −3.99550 + 0.189670i 0.521078 + 0.437236i −0.520461 + 3.42478i −2.27989 + 1.31629i 0.568744 + 7.97976i 2.81908 + 1.02606i 0.849512 − 1.06260i
43.12 0.516673 − 1.93211i −1.70574 − 0.300767i −3.46610 − 1.99654i 5.90515 + 4.95501i −1.46242 + 3.14027i 1.82154 − 1.05167i −5.64837 + 5.66533i 2.81908 + 1.02606i 12.6246 − 8.84928i
43.13 0.541457 + 1.92531i −1.70574 − 0.300767i −3.41365 + 2.08495i −0.464266 − 0.389565i −0.344512 − 3.44693i −1.87243 + 1.08105i −5.86252 − 5.44343i 2.81908 + 1.02606i 0.498654 − 1.10479i
43.14 1.15879 − 1.63009i −1.70574 − 0.300767i −1.31441 − 3.77787i −0.355903 − 0.298638i −2.46687 + 2.43198i 4.94187 − 2.85319i −7.68141 − 2.23515i 2.81908 + 1.02606i −0.899225 + 0.234097i
43.15 1.21874 + 1.58577i −1.70574 − 0.300767i −1.02936 + 3.86528i 5.28226 + 4.43234i −1.60189 − 3.07147i 10.8274 − 6.25118i −7.38399 + 3.07842i 2.81908 + 1.02606i −0.591014 + 13.7783i
43.16 1.30280 − 1.51747i −1.70574 − 0.300767i −0.605405 − 3.95392i −2.99833 − 2.51590i −2.67864 + 2.19656i −10.3974 + 6.00295i −6.78866 − 4.23250i 2.81908 + 1.02606i −7.72404 + 1.27214i
43.17 1.82571 + 0.816563i −1.70574 − 0.300767i 2.66645 + 2.98162i 2.82580 + 2.37113i −2.86859 − 1.94196i −8.85535 + 5.11264i 2.43349 + 7.62090i 2.81908 + 1.02606i 3.22292 + 6.63644i
43.18 1.86747 + 0.715933i −1.70574 − 0.300767i 2.97488 + 2.67396i −3.79119 − 3.18119i −2.97008 − 1.78287i 8.30684 − 4.79596i 3.64112 + 7.12336i 2.81908 + 1.02606i −4.80242 − 8.65501i
43.19 1.90278 − 0.615983i −1.70574 − 0.300767i 3.24113 − 2.34416i 3.87292 + 3.24977i −3.43091 + 0.478412i 1.81496 − 1.04787i 4.72318 − 6.45690i 2.81908 + 1.02606i 9.37112 + 3.79793i
43.20 1.96554 − 0.369657i −1.70574 − 0.300767i 3.72671 − 1.45315i −5.33708 − 4.47835i −3.46388 + 0.0393662i −1.82806 + 1.05543i 6.78783 − 4.23383i 2.81908 + 1.02606i −12.1457 − 6.82948i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.l 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.x.a ✓ 120
4.b odd 2 1 228.3.x.b yes 120
19.e even 9 1 228.3.x.b yes 120
76.l odd 18 1 inner 228.3.x.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.x.a ✓ 120 1.a even 1 1 trivial
228.3.x.a ✓ 120 76.l odd 18 1 inner
228.3.x.b yes 120 4.b odd 2 1
228.3.x.b yes 120 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{120} - 1680 T_{7}^{118} + 1501518 T_{7}^{116} + 171756 T_{7}^{115} - 925953068 T_{7}^{114} + \cdots + 46\!\cdots\!09 \) acting on \(S_{3}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display