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Properties

Label 333.3.b.a
Level $333$
Weight $3$
Character orbit 333.b
Analytic conductor $9.074$
Analytic rank $0$
Dimension $24$
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(260,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.260");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.b (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 40 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 40 q^{4} - 16 q^{7} + 8 q^{10} + 8 q^{13} + 40 q^{16} + 24 q^{19} + 72 q^{22} - 112 q^{25} - 8 q^{28} - 48 q^{31} + 8 q^{34} - 200 q^{40} + 16 q^{43} + 256 q^{46} + 368 q^{49} - 280 q^{52} - 96 q^{55} - 80 q^{58} + 256 q^{64} - 152 q^{67} + 360 q^{70} - 448 q^{73} - 440 q^{76} + 312 q^{79} - 152 q^{82} + 168 q^{85} + 32 q^{88} + 168 q^{91} + 336 q^{94} + 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
260.1 − 3.55391i 0 −8.63025 5.62118i 0 −3.90000 16.4555i 0 19.9771
260.2 − 3.50432i 0 −8.28023 3.74537i 0 12.4695 14.9993i 0 13.1250
260.3 − 3.44796i 0 −7.88842 − 4.26956i 0 −6.17353 13.4071i 0 −14.7213
260.4 − 3.13452i 0 −5.82522 − 2.26469i 0 3.42555 5.72120i 0 −7.09872
260.5 − 2.45758i 0 −2.03969 − 8.98168i 0 −12.0137 − 4.81762i 0 −22.0732
260.6 − 2.42520i 0 −1.88158 7.03447i 0 −7.08791 − 5.13759i 0 17.0600
260.7 − 1.81207i 0 0.716386 1.64799i 0 8.99296 − 8.54644i 0 2.98627
260.8 − 1.77587i 0 0.846301 0.197521i 0 −1.70982 − 8.60638i 0 0.350771
260.9 − 1.30759i 0 2.29021 − 8.83544i 0 6.87587 − 8.22501i 0 −11.5531
260.10 − 0.995674i 0 3.00863 4.93961i 0 −13.0474 − 6.97831i 0 4.91824
260.11 − 0.559794i 0 3.68663 1.17165i 0 −2.75783 − 4.30293i 0 0.655884
260.12 − 0.0526181i 0 3.99723 7.09012i 0 6.92631 − 0.420800i 0 0.373069
260.13 0.0526181i 0 3.99723 − 7.09012i 0 6.92631 0.420800i 0 0.373069
260.14 0.559794i 0 3.68663 − 1.17165i 0 −2.75783 4.30293i 0 0.655884
260.15 0.995674i 0 3.00863 − 4.93961i 0 −13.0474 6.97831i 0 4.91824
260.16 1.30759i 0 2.29021 8.83544i 0 6.87587 8.22501i 0 −11.5531
260.17 1.77587i 0 0.846301 − 0.197521i 0 −1.70982 8.60638i 0 0.350771
260.18 1.81207i 0 0.716386 − 1.64799i 0 8.99296 8.54644i 0 2.98627
260.19 2.42520i 0 −1.88158 − 7.03447i 0 −7.08791 5.13759i 0 17.0600
260.20 2.45758i 0 −2.03969 8.98168i 0 −12.0137 4.81762i 0 −22.0732
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 260.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.3.b.a ✓ 24
3.b odd 2 1 inner 333.3.b.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
333.3.b.a ✓ 24 1.a even 1 1 trivial
333.3.b.a ✓ 24 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(333, [\chi])\).