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Properties

Label 333.3.bu.e
Level $333$
Weight $3$
Character orbit 333.bu
Analytic conductor $9.074$
Analytic rank $0$
Dimension $144$
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(19,333)]
 
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([0, 35]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.bu (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: \(9.07359280320\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 36 q^{4} - 12 q^{16} - 144 q^{19} - 252 q^{25} - 336 q^{28} + 24 q^{31} + 180 q^{34} - 588 q^{37} + 144 q^{40} + 456 q^{43} + 180 q^{46} + 720 q^{49} - 120 q^{52} - 108 q^{55} + 204 q^{58} + 432 q^{61} + 2268 q^{64} + 456 q^{67} + 1128 q^{70} - 684 q^{76} - 432 q^{79} - 900 q^{82} - 1944 q^{85} - 1656 q^{88} - 2484 q^{91} - 576 q^{94} - 1116 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.89678 + 0.340924i 0 11.1294 − 1.96241i −2.27531 − 4.87941i 0 0.0236235 + 0.00859825i −27.5862 + 7.39171i 0 10.5299 + 18.2383i
19.2 −3.43199 + 0.300260i 0 7.74918 − 1.36639i 3.57201 + 7.66021i 0 −6.17371 − 2.24705i −12.8740 + 3.44958i 0 −14.5592 − 25.2172i
19.3 −2.72879 + 0.238738i 0 3.45005 − 0.608338i −0.371663 − 0.797033i 0 10.3536 + 3.76841i 1.31427 − 0.352157i 0 1.20447 + 2.08620i
19.4 −2.22054 + 0.194272i 0 0.953835 − 0.168187i −3.70643 − 7.94846i 0 −10.2524 − 3.73158i 6.52693 − 1.74889i 0 9.77445 + 16.9298i
19.5 −1.89218 + 0.165545i 0 −0.386275 + 0.0681108i 0.267317 + 0.573262i 0 −5.20660 − 1.89505i 8.05839 − 2.15924i 0 −0.600713 − 1.04047i
19.6 −0.776344 + 0.0679213i 0 −3.34113 + 0.589132i 3.69628 + 7.92669i 0 5.62547 + 2.04750i 5.56487 − 1.49110i 0 −3.40797 − 5.90278i
19.7 0.776344 − 0.0679213i 0 −3.34113 + 0.589132i −3.69628 − 7.92669i 0 5.62547 + 2.04750i −5.56487 + 1.49110i 0 −3.40797 − 5.90278i
19.8 1.89218 − 0.165545i 0 −0.386275 + 0.0681108i −0.267317 − 0.573262i 0 −5.20660 − 1.89505i −8.05839 + 2.15924i 0 −0.600713 − 1.04047i
19.9 2.22054 − 0.194272i 0 0.953835 − 0.168187i 3.70643 + 7.94846i 0 −10.2524 − 3.73158i −6.52693 + 1.74889i 0 9.77445 + 16.9298i
19.10 2.72879 − 0.238738i 0 3.45005 − 0.608338i 0.371663 + 0.797033i 0 10.3536 + 3.76841i −1.31427 + 0.352157i 0 1.20447 + 2.08620i
19.11 3.43199 − 0.300260i 0 7.74918 − 1.36639i −3.57201 − 7.66021i 0 −6.17371 − 2.24705i 12.8740 − 3.44958i 0 −14.5592 − 25.2172i
19.12 3.89678 − 0.340924i 0 11.1294 − 1.96241i 2.27531 + 4.87941i 0 0.0236235 + 0.00859825i 27.5862 − 7.39171i 0 10.5299 + 18.2383i
55.1 −0.319790 − 3.65521i 0 −9.31908 + 1.64321i −3.65685 + 1.70522i 0 −4.54508 − 1.65427i 5.18780 + 19.3611i 0 7.40235 + 12.8212i
55.2 −0.277211 − 3.16853i 0 −6.02352 + 1.06211i 0.157508 − 0.0734471i 0 11.7519 + 4.27736i 1.74227 + 6.50225i 0 −0.276382 − 0.478708i
55.3 −0.213166 − 2.43650i 0 −1.95188 + 0.344168i 6.45535 − 3.01018i 0 2.88795 + 1.05113i −1.27745 − 4.76750i 0 −8.71037 − 15.0868i
55.4 −0.195550 − 2.23514i 0 −1.01839 + 0.179569i −2.56773 + 1.19735i 0 −4.76529 − 1.73443i −1.72232 − 6.42778i 0 3.17837 + 5.50510i
55.5 −0.0825401 − 0.943438i 0 3.05597 − 0.538850i 5.72677 − 2.67044i 0 −9.29236 − 3.38214i −1.74106 − 6.49773i 0 −2.99208 − 5.18243i
55.6 −0.0703221 − 0.803785i 0 3.29811 − 0.581545i −6.85966 + 3.19871i 0 9.59287 + 3.49152i −1.53469 − 5.72752i 0 3.05346 + 5.28875i
55.7 0.0703221 + 0.803785i 0 3.29811 − 0.581545i 6.85966 − 3.19871i 0 9.59287 + 3.49152i 1.53469 + 5.72752i 0 3.05346 + 5.28875i
55.8 0.0825401 + 0.943438i 0 3.05597 − 0.538850i −5.72677 + 2.67044i 0 −9.29236 − 3.38214i 1.74106 + 6.49773i 0 −2.99208 − 5.18243i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
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 333.3.bu.e ✓ 144
3.b odd 2 1 inner 333.3.bu.e ✓ 144
37.i odd 36 1 inner 333.3.bu.e ✓ 144
111.q even 36 1 inner 333.3.bu.e ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
333.3.bu.e ✓ 144 1.a even 1 1 trivial
333.3.bu.e ✓ 144 3.b odd 2 1 inner
333.3.bu.e ✓ 144 37.i odd 36 1 inner
333.3.bu.e ✓ 144 111.q even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - 18 T_{2}^{142} + 165 T_{2}^{140} - 1404 T_{2}^{138} + 10662 T_{2}^{136} + \cdots + 12\!\cdots\!29 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display