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Properties

Label 333.3.bb.d
Level $333$
Weight $3$
Character orbit 333.bb
Analytic conductor $9.074$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [333,3,Mod(82,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.bb (of order \(12\), degree \(4\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 36 q^{4} + 24 q^{10} - 40 q^{13} + 52 q^{16} + 120 q^{19} + 40 q^{22} + 252 q^{25} - 96 q^{28} + 24 q^{31} - 28 q^{34} + 184 q^{37} + 72 q^{40} - 196 q^{43} + 300 q^{46} - 184 q^{49} - 88 q^{52} - 164 q^{55} + 336 q^{58} + 504 q^{61} - 24 q^{67} - 156 q^{70} + 156 q^{76} - 472 q^{79} - 868 q^{82} - 596 q^{88} + 328 q^{91} - 40 q^{94} + 44 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −3.39876 − 0.910696i 0 7.25813 + 4.19048i −2.94261 + 0.788470i 0 0.170998 − 0.296177i −10.9001 − 10.9001i 0 10.7193
82.2 −2.62415 − 0.703138i 0 2.92764 + 1.69027i 4.96602 − 1.33064i 0 −5.48495 + 9.50021i 1.18998 + 1.18998i 0 −13.9672
82.3 −2.13643 − 0.572455i 0 0.772537 + 0.446024i 4.15376 − 1.11300i 0 4.90291 − 8.49210i 4.86076 + 4.86076i 0 −9.51137
82.4 −1.66731 − 0.446755i 0 −0.883758 − 0.510238i −9.59098 + 2.56990i 0 2.12740 − 3.68476i 6.12778 + 6.12778i 0 17.1393
82.5 −1.02941 − 0.275830i 0 −2.48049 − 1.43211i 0.765915 − 0.205226i 0 −0.177036 + 0.306635i 5.17276 + 5.17276i 0 −0.845051
82.6 −0.180695 − 0.0484170i 0 −3.43380 − 1.98250i −6.18101 + 1.65620i 0 −3.27138 + 5.66619i 1.05359 + 1.05359i 0 1.19706
82.7 0.180695 + 0.0484170i 0 −3.43380 − 1.98250i 6.18101 − 1.65620i 0 −3.27138 + 5.66619i −1.05359 − 1.05359i 0 1.19706
82.8 1.02941 + 0.275830i 0 −2.48049 − 1.43211i −0.765915 + 0.205226i 0 −0.177036 + 0.306635i −5.17276 − 5.17276i 0 −0.845051
82.9 1.66731 + 0.446755i 0 −0.883758 − 0.510238i 9.59098 − 2.56990i 0 2.12740 − 3.68476i −6.12778 − 6.12778i 0 17.1393
82.10 2.13643 + 0.572455i 0 0.772537 + 0.446024i −4.15376 + 1.11300i 0 4.90291 − 8.49210i −4.86076 − 4.86076i 0 −9.51137
82.11 2.62415 + 0.703138i 0 2.92764 + 1.69027i −4.96602 + 1.33064i 0 −5.48495 + 9.50021i −1.18998 − 1.18998i 0 −13.9672
82.12 3.39876 + 0.910696i 0 7.25813 + 4.19048i 2.94261 − 0.788470i 0 0.170998 − 0.296177i 10.9001 + 10.9001i 0 10.7193
199.1 −3.39876 + 0.910696i 0 7.25813 − 4.19048i −2.94261 − 0.788470i 0 0.170998 + 0.296177i −10.9001 + 10.9001i 0 10.7193
199.2 −2.62415 + 0.703138i 0 2.92764 − 1.69027i 4.96602 + 1.33064i 0 −5.48495 − 9.50021i 1.18998 − 1.18998i 0 −13.9672
199.3 −2.13643 + 0.572455i 0 0.772537 − 0.446024i 4.15376 + 1.11300i 0 4.90291 + 8.49210i 4.86076 − 4.86076i 0 −9.51137
199.4 −1.66731 + 0.446755i 0 −0.883758 + 0.510238i −9.59098 − 2.56990i 0 2.12740 + 3.68476i 6.12778 − 6.12778i 0 17.1393
199.5 −1.02941 + 0.275830i 0 −2.48049 + 1.43211i 0.765915 + 0.205226i 0 −0.177036 − 0.306635i 5.17276 − 5.17276i 0 −0.845051
199.6 −0.180695 + 0.0484170i 0 −3.43380 + 1.98250i −6.18101 − 1.65620i 0 −3.27138 − 5.66619i 1.05359 − 1.05359i 0 1.19706
199.7 0.180695 − 0.0484170i 0 −3.43380 + 1.98250i 6.18101 + 1.65620i 0 −3.27138 − 5.66619i −1.05359 + 1.05359i 0 1.19706
199.8 1.02941 − 0.275830i 0 −2.48049 + 1.43211i −0.765915 − 0.205226i 0 −0.177036 − 0.306635i −5.17276 + 5.17276i 0 −0.845051
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.g odd 12 1 inner
111.m even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.3.bb.d ✓ 48
3.b odd 2 1 inner 333.3.bb.d ✓ 48
37.g odd 12 1 inner 333.3.bb.d ✓ 48
111.m even 12 1 inner 333.3.bb.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
333.3.bb.d ✓ 48 1.a even 1 1 trivial
333.3.bb.d ✓ 48 3.b odd 2 1 inner
333.3.bb.d ✓ 48 37.g odd 12 1 inner
333.3.bb.d ✓ 48 111.m even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 18 T_{2}^{46} - 163 T_{2}^{44} - 4878 T_{2}^{42} + 26430 T_{2}^{40} + 884730 T_{2}^{38} + \cdots + 27439591201 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display