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Properties

Label 333.3.bb.b
Level $333$
Weight $3$
Character orbit 333.bb
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(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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 111)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 18 q^{4} - 8 q^{5} + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 18 q^{4} - 8 q^{5} + 36 q^{8} - 60 q^{10} + 28 q^{13} - 42 q^{14} + 26 q^{16} - 10 q^{17} + 60 q^{19} + 4 q^{20} - 64 q^{22} - 34 q^{23} - 162 q^{25} + 44 q^{26} - 48 q^{28} - 32 q^{29} + 90 q^{32} + 46 q^{34} + 30 q^{35} + 80 q^{37} + 284 q^{38} - 144 q^{40} + 30 q^{41} + 130 q^{43} + 16 q^{44} + 78 q^{46} + 56 q^{47} - 20 q^{49} - 70 q^{50} + 16 q^{52} + 190 q^{53} + 350 q^{55} - 376 q^{56} + 336 q^{58} + 258 q^{59} - 84 q^{61} + 474 q^{62} + 54 q^{65} - 372 q^{67} + 434 q^{68} + 102 q^{70} - 66 q^{71} + 416 q^{74} + 702 q^{76} - 198 q^{77} + 88 q^{79} - 900 q^{80} - 470 q^{82} - 166 q^{83} - 432 q^{86} + 530 q^{88} - 304 q^{89} + 524 q^{91} - 330 q^{92} - 344 q^{94} - 1080 q^{95} - 110 q^{97} - 926 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −3.09488 − 0.829271i 0 5.42650 + 3.13299i 3.56389 − 0.954941i 0 −1.12003 + 1.93996i −5.13384 − 5.13384i 0 −11.8217
82.2 −2.32451 − 0.622850i 0 1.55129 + 0.895636i −3.55969 + 0.953815i 0 1.46050 − 2.52967i 3.75850 + 3.75850i 0 8.86860
82.3 0.118812 + 0.0318357i 0 −3.45100 − 1.99244i −7.22425 + 1.93573i 0 0.350980 − 0.607914i −0.694498 − 0.694498i 0 −0.919956
82.4 0.839390 + 0.224914i 0 −2.81011 − 1.62242i 2.87694 − 0.770875i 0 −5.46857 + 9.47184i −4.45178 − 4.45178i 0 2.58826
82.5 2.51039 + 0.672656i 0 2.38548 + 1.37726i 2.86514 − 0.767711i 0 4.00759 − 6.94135i −2.28887 − 2.28887i 0 7.70901
82.6 3.81682 + 1.02271i 0 10.0581 + 5.80704i −3.12011 + 0.836031i 0 −4.42663 + 7.66714i 21.2746 + 21.2746i 0 −12.7639
199.1 −3.09488 + 0.829271i 0 5.42650 − 3.13299i 3.56389 + 0.954941i 0 −1.12003 − 1.93996i −5.13384 + 5.13384i 0 −11.8217
199.2 −2.32451 + 0.622850i 0 1.55129 − 0.895636i −3.55969 − 0.953815i 0 1.46050 + 2.52967i 3.75850 − 3.75850i 0 8.86860
199.3 0.118812 − 0.0318357i 0 −3.45100 + 1.99244i −7.22425 − 1.93573i 0 0.350980 + 0.607914i −0.694498 + 0.694498i 0 −0.919956
199.4 0.839390 − 0.224914i 0 −2.81011 + 1.62242i 2.87694 + 0.770875i 0 −5.46857 − 9.47184i −4.45178 + 4.45178i 0 2.58826
199.5 2.51039 − 0.672656i 0 2.38548 − 1.37726i 2.86514 + 0.767711i 0 4.00759 + 6.94135i −2.28887 + 2.28887i 0 7.70901
199.6 3.81682 − 1.02271i 0 10.0581 − 5.80704i −3.12011 − 0.836031i 0 −4.42663 − 7.66714i 21.2746 − 21.2746i 0 −12.7639
208.1 −0.821656 − 3.06646i 0 −5.26396 + 3.03915i 1.49867 − 5.59311i 0 −0.749067 − 1.29742i 4.66537 + 4.66537i 0 −18.3824
208.2 −0.544686 − 2.03280i 0 −0.371475 + 0.214471i −1.21483 + 4.53380i 0 1.71728 + 2.97442i −5.31413 − 5.31413i 0 9.87799
208.3 0.0210891 + 0.0787054i 0 3.45835 − 1.99668i −0.222135 + 0.829019i 0 −2.82588 − 4.89456i 0.460548 + 0.460548i 0 −0.0699329
208.4 0.103369 + 0.385779i 0 3.32596 − 1.92024i 2.38266 − 8.89221i 0 6.18137 + 10.7065i 2.21423 + 2.21423i 0 3.67672
208.5 0.682073 + 2.54553i 0 −2.55040 + 1.47247i −2.05541 + 7.67090i 0 4.48853 + 7.77437i 1.96605 + 1.96605i 0 −20.9284
208.6 0.693785 + 2.58924i 0 −2.75873 + 1.59276i 0.209120 − 0.780447i 0 −3.61609 − 6.26326i 1.54383 + 1.54383i 0 2.16585
325.1 −0.821656 + 3.06646i 0 −5.26396 − 3.03915i 1.49867 + 5.59311i 0 −0.749067 + 1.29742i 4.66537 − 4.66537i 0 −18.3824
325.2 −0.544686 + 2.03280i 0 −0.371475 − 0.214471i −1.21483 − 4.53380i 0 1.71728 − 2.97442i −5.31413 + 5.31413i 0 9.87799
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 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.b 24
3.b odd 2 1 111.3.l.b ✓ 24
37.g odd 12 1 inner 333.3.bb.b 24
111.m even 12 1 111.3.l.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.l.b ✓ 24 3.b odd 2 1
111.3.l.b ✓ 24 111.m even 12 1
333.3.bb.b 24 1.a even 1 1 trivial
333.3.bb.b 24 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 4 T_{2}^{23} - T_{2}^{22} + 8 T_{2}^{21} - 114 T_{2}^{20} + 562 T_{2}^{19} + 109 T_{2}^{18} + \cdots + 169 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display