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Properties

Label 333.3.i.b
Level $333$
Weight $3$
Character orbit 333.i
Analytic conductor $9.074$
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] = [333,3,Mod(154,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.154");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.i (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: \(9.07359280320\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 111)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} - 20 q^{5} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} - 20 q^{5} + 60 q^{8} + 24 q^{10} - 32 q^{13} - 60 q^{14} - 40 q^{16} + 8 q^{17} - 48 q^{19} - 104 q^{20} + 32 q^{22} + 44 q^{23} - 88 q^{26} + 88 q^{29} + 192 q^{32} - 56 q^{34} - 60 q^{35} + 56 q^{37} - 160 q^{38} - 8 q^{43} - 536 q^{44} + 480 q^{46} - 88 q^{47} - 32 q^{49} - 172 q^{50} - 488 q^{52} - 104 q^{53} - 208 q^{55} + 80 q^{56} + 456 q^{59} - 96 q^{61} + 44 q^{68} - 1080 q^{70} + 528 q^{71} - 136 q^{74} - 24 q^{76} + 160 q^{79} - 48 q^{80} - 392 q^{82} - 280 q^{83} + 288 q^{86} + 1424 q^{88} + 92 q^{89} + 320 q^{91} - 48 q^{92} - 56 q^{94} + 352 q^{97} + 664 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
154.1 −2.61750 + 2.61750i 0 − 9.70258i −5.94940 − 5.94940i 0 −5.80748 14.9265 + 14.9265i 0 31.1451
154.2 −2.61593 + 2.61593i 0 − 9.68620i 2.59767 + 2.59767i 0 8.77260 14.8747 + 14.8747i 0 −13.5907
154.3 −2.00072 + 2.00072i 0 − 4.00579i 3.43593 + 3.43593i 0 2.85072 0.0115823 + 0.0115823i 0 −13.7487
154.4 −1.81870 + 1.81870i 0 − 2.61536i −5.47037 − 5.47037i 0 −2.07266 −2.51825 − 2.51825i 0 19.8979
154.5 −1.06504 + 1.06504i 0 1.73137i 0.563895 + 0.563895i 0 6.70948 −6.10415 − 6.10415i 0 −1.20114
154.6 −0.746615 + 0.746615i 0 2.88513i −3.97615 − 3.97615i 0 4.59754 −5.14054 − 5.14054i 0 5.93731
154.7 0.123529 − 0.123529i 0 3.96948i −1.06915 − 1.06915i 0 −7.73043 0.984460 + 0.984460i 0 −0.264141
154.8 0.284993 − 0.284993i 0 3.83756i 5.88958 + 5.88958i 0 −7.94417 2.23365 + 2.23365i 0 3.35698
154.9 0.921681 − 0.921681i 0 2.30101i −2.11453 − 2.11453i 0 3.42289 5.80752 + 5.80752i 0 −3.89784
154.10 1.67294 − 1.67294i 0 − 1.59747i 4.04481 + 4.04481i 0 −2.77594 4.01929 + 4.01929i 0 13.5335
154.11 1.70321 − 1.70321i 0 − 1.80182i −5.66686 − 5.66686i 0 10.9534 3.74395 + 3.74395i 0 −19.3037
154.12 2.15816 − 2.15816i 0 − 5.31533i −2.28543 − 2.28543i 0 −10.9759 −2.83869 − 2.83869i 0 −9.86464
253.1 −2.61750 − 2.61750i 0 9.70258i −5.94940 + 5.94940i 0 −5.80748 14.9265 − 14.9265i 0 31.1451
253.2 −2.61593 − 2.61593i 0 9.68620i 2.59767 − 2.59767i 0 8.77260 14.8747 − 14.8747i 0 −13.5907
253.3 −2.00072 − 2.00072i 0 4.00579i 3.43593 − 3.43593i 0 2.85072 0.0115823 − 0.0115823i 0 −13.7487
253.4 −1.81870 − 1.81870i 0 2.61536i −5.47037 + 5.47037i 0 −2.07266 −2.51825 + 2.51825i 0 19.8979
253.5 −1.06504 − 1.06504i 0 − 1.73137i 0.563895 − 0.563895i 0 6.70948 −6.10415 + 6.10415i 0 −1.20114
253.6 −0.746615 − 0.746615i 0 − 2.88513i −3.97615 + 3.97615i 0 4.59754 −5.14054 + 5.14054i 0 5.93731
253.7 0.123529 + 0.123529i 0 − 3.96948i −1.06915 + 1.06915i 0 −7.73043 0.984460 − 0.984460i 0 −0.264141
253.8 0.284993 + 0.284993i 0 − 3.83756i 5.88958 − 5.88958i 0 −7.94417 2.23365 − 2.23365i 0 3.35698
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 154.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.3.i.b 24
3.b odd 2 1 111.3.f.a ✓ 24
37.d odd 4 1 inner 333.3.i.b 24
111.g even 4 1 111.3.f.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.f.a ✓ 24 3.b odd 2 1
111.3.f.a ✓ 24 111.g even 4 1
333.3.i.b 24 1.a even 1 1 trivial
333.3.i.b 24 37.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 8 T_{2}^{23} + 32 T_{2}^{22} + 44 T_{2}^{21} + 162 T_{2}^{20} + 1168 T_{2}^{19} + \cdots + 64009 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display