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Properties

Label 333.2.br.a
Level $333$
Weight $2$
Character orbit 333.br
Analytic conductor $2.659$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [333,2,Mod(17,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 333.br (of order \(36\), degree \(12\), 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: \(2.65901838731\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(12\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 24 q^{16} - 48 q^{28} - 48 q^{31} - 144 q^{34} - 12 q^{37} - 72 q^{40} - 48 q^{43} - 216 q^{46} + 108 q^{49} - 120 q^{52} + 132 q^{58} - 48 q^{67} + 48 q^{70} + 72 q^{82} + 144 q^{88} - 108 q^{91} + 144 q^{94} + 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.33459 + 1.90599i 0 −1.16763 − 3.20804i −2.39882 + 0.209870i 0 −0.598713 − 0.502380i 3.17779 + 0.851487i 0 2.80143 − 4.85222i
17.2 −1.32779 + 1.89628i 0 −1.14882 − 3.15635i 2.61557 − 0.228833i 0 −3.46230 − 2.90521i 3.03863 + 0.814197i 0 −3.03900 + 5.26370i
17.3 −1.16094 + 1.65800i 0 −0.717130 − 1.97030i −1.09531 + 0.0958273i 0 3.14578 + 2.63962i 0.189150 + 0.0506825i 0 1.11271 − 1.92727i
17.4 −0.669250 + 0.955789i 0 0.218405 + 0.600062i 1.25001 − 0.109362i 0 1.89189 + 1.58748i −2.97379 − 0.796824i 0 −0.732043 + 1.26794i
17.5 −0.499615 + 0.713524i 0 0.424538 + 1.16641i −2.38893 + 0.209004i 0 −1.79627 − 1.50725i −2.72711 − 0.730727i 0 1.04442 − 1.80898i
17.6 −0.0580578 + 0.0829151i 0 0.680536 + 1.86976i 3.99008 − 0.349086i 0 0.295606 + 0.248043i −0.390085 − 0.104523i 0 −0.202710 + 0.351105i
17.7 0.0580578 − 0.0829151i 0 0.680536 + 1.86976i −3.99008 + 0.349086i 0 0.295606 + 0.248043i 0.390085 + 0.104523i 0 −0.202710 + 0.351105i
17.8 0.499615 − 0.713524i 0 0.424538 + 1.16641i 2.38893 − 0.209004i 0 −1.79627 − 1.50725i 2.72711 + 0.730727i 0 1.04442 − 1.80898i
17.9 0.669250 − 0.955789i 0 0.218405 + 0.600062i −1.25001 + 0.109362i 0 1.89189 + 1.58748i 2.97379 + 0.796824i 0 −0.732043 + 1.26794i
17.10 1.16094 − 1.65800i 0 −0.717130 − 1.97030i 1.09531 − 0.0958273i 0 3.14578 + 2.63962i −0.189150 − 0.0506825i 0 1.11271 − 1.92727i
17.11 1.32779 − 1.89628i 0 −1.14882 − 3.15635i −2.61557 + 0.228833i 0 −3.46230 − 2.90521i −3.03863 − 0.814197i 0 −3.03900 + 5.26370i
17.12 1.33459 − 1.90599i 0 −1.16763 − 3.20804i 2.39882 − 0.209870i 0 −0.598713 − 0.502380i −3.17779 − 0.851487i 0 2.80143 − 4.85222i
35.1 −2.46829 − 0.215947i 0 4.07619 + 0.718743i −1.12826 + 2.41956i 0 −3.45814 + 1.25866i −5.11942 − 1.37174i 0 3.30737 − 5.72853i
35.2 −2.22146 − 0.194353i 0 2.92750 + 0.516197i 1.38175 − 2.96317i 0 3.02010 − 1.09923i −2.09507 − 0.561372i 0 −3.64540 + 6.31402i
35.3 −2.04543 − 0.178952i 0 2.18213 + 0.384768i −0.900528 + 1.93119i 0 1.38265 − 0.503244i −0.427968 − 0.114674i 0 2.18755 − 3.78895i
35.4 −1.04267 − 0.0912222i 0 −0.890767 − 0.157066i 1.62802 − 3.49130i 0 −0.232992 + 0.0848021i 2.93644 + 0.786816i 0 −2.01598 + 3.49178i
35.5 −0.740607 − 0.0647947i 0 −1.42532 − 0.251322i 0.605949 − 1.29946i 0 −3.90763 + 1.42226i 2.47552 + 0.663314i 0 −0.532968 + 0.923127i
35.6 −0.155263 − 0.0135838i 0 −1.94569 − 0.343078i −0.307182 + 0.658754i 0 1.34518 − 0.489605i 0.598525 + 0.160374i 0 0.0566423 − 0.0981074i
35.7 0.155263 + 0.0135838i 0 −1.94569 − 0.343078i 0.307182 − 0.658754i 0 1.34518 − 0.489605i −0.598525 − 0.160374i 0 0.0566423 − 0.0981074i
35.8 0.740607 + 0.0647947i 0 −1.42532 − 0.251322i −0.605949 + 1.29946i 0 −3.90763 + 1.42226i −2.47552 − 0.663314i 0 −0.532968 + 0.923127i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.i odd 36 1 inner
111.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.2.br.a ✓ 144
3.b odd 2 1 inner 333.2.br.a ✓ 144
37.i odd 36 1 inner 333.2.br.a ✓ 144
111.q even 36 1 inner 333.2.br.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
333.2.br.a ✓ 144 1.a even 1 1 trivial
333.2.br.a ✓ 144 3.b odd 2 1 inner
333.2.br.a ✓ 144 37.i odd 36 1 inner
333.2.br.a ✓ 144 111.q even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(333, [\chi])\).