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Properties

Label 333.3.bb.a
Level $333$
Weight $3$
Character orbit 333.bb
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(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 37)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 18 q^{5} - 2 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} + 18 q^{5} - 2 q^{7} - 24 q^{8} + 4 q^{10} - 20 q^{13} - 16 q^{14} + 60 q^{16} + 32 q^{17} + 52 q^{19} + 90 q^{20} - 4 q^{22} + 8 q^{23} + 120 q^{25} - 96 q^{26} + 138 q^{28} + 24 q^{29} - 72 q^{31} - 152 q^{32} - 64 q^{34} - 10 q^{35} - 66 q^{37} - 384 q^{38} - 282 q^{40} + 48 q^{41} - 304 q^{43} + 392 q^{44} - 122 q^{46} - 240 q^{47} - 12 q^{49} + 356 q^{50} + 332 q^{52} - 32 q^{53} + 128 q^{55} - 506 q^{56} - 198 q^{58} - 296 q^{59} - 246 q^{61} - 540 q^{62} + 60 q^{65} + 224 q^{68} - 322 q^{70} + 218 q^{71} - 520 q^{74} - 132 q^{76} + 144 q^{77} + 284 q^{79} + 1576 q^{80} - 308 q^{82} + 54 q^{83} + 182 q^{86} + 212 q^{88} + 128 q^{89} - 468 q^{91} - 386 q^{92} - 148 q^{94} + 220 q^{97} + 452 q^{98}+O(q^{100}) \) 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 −3.39368 − 0.909333i 0 7.22605 + 4.17196i 6.92898 − 1.85661i 0 3.98367 − 6.89992i −10.7918 − 10.7918i 0 −25.2030
82.2 −1.77110 − 0.474565i 0 −0.552513 − 0.318994i −2.85393 + 0.764709i 0 −4.92129 + 8.52393i 6.01332 + 6.01332i 0 5.41751
82.3 −0.534561 − 0.143235i 0 −3.19886 − 1.84686i 6.19577 − 1.66015i 0 5.09607 − 8.82666i 3.01075 + 3.01075i 0 −3.54981
82.4 0.474550 + 0.127155i 0 −3.25507 − 1.87932i −1.16277 + 0.311563i 0 1.65463 − 2.86590i −2.69531 − 2.69531i 0 −0.591409
82.5 2.31542 + 0.620415i 0 1.51216 + 0.873047i −7.54635 + 2.02204i 0 1.46125 − 2.53095i −3.82039 − 3.82039i 0 −18.7275
82.6 2.90937 + 0.779562i 0 4.39259 + 2.53606i 9.00048 − 2.41167i 0 −3.44420 + 5.96552i 2.28342 + 2.28342i 0 28.0657
199.1 −3.39368 + 0.909333i 0 7.22605 − 4.17196i 6.92898 + 1.85661i 0 3.98367 + 6.89992i −10.7918 + 10.7918i 0 −25.2030
199.2 −1.77110 + 0.474565i 0 −0.552513 + 0.318994i −2.85393 − 0.764709i 0 −4.92129 − 8.52393i 6.01332 − 6.01332i 0 5.41751
199.3 −0.534561 + 0.143235i 0 −3.19886 + 1.84686i 6.19577 + 1.66015i 0 5.09607 + 8.82666i 3.01075 − 3.01075i 0 −3.54981
199.4 0.474550 − 0.127155i 0 −3.25507 + 1.87932i −1.16277 − 0.311563i 0 1.65463 + 2.86590i −2.69531 + 2.69531i 0 −0.591409
199.5 2.31542 − 0.620415i 0 1.51216 − 0.873047i −7.54635 − 2.02204i 0 1.46125 + 2.53095i −3.82039 + 3.82039i 0 −18.7275
199.6 2.90937 − 0.779562i 0 4.39259 − 2.53606i 9.00048 + 2.41167i 0 −3.44420 − 5.96552i 2.28342 − 2.28342i 0 28.0657
208.1 −0.983199 − 3.66935i 0 −9.03333 + 5.21539i −2.06524 + 7.70758i 0 −0.830601 − 1.43864i 17.2741 + 17.2741i 0 30.3123
208.2 −0.540554 − 2.01737i 0 −0.313497 + 0.180998i −0.199493 + 0.744517i 0 4.01523 + 6.95457i −5.37268 − 5.37268i 0 1.60981
208.3 −0.438633 − 1.63700i 0 0.976729 − 0.563915i 1.36471 − 5.09318i 0 −5.80200 − 10.0494i −6.14503 − 6.14503i 0 −8.93615
208.4 0.243224 + 0.907724i 0 2.69930 − 1.55844i −0.778419 + 2.90510i 0 1.69522 + 2.93621i 4.72917 + 4.72917i 0 −2.82636
208.5 0.756076 + 2.82171i 0 −3.92632 + 2.26686i 1.69651 − 6.33147i 0 −0.190362 − 0.329717i −1.10248 − 1.10248i 0 19.1483
208.6 0.963085 + 3.59428i 0 −8.52723 + 4.92320i −1.58025 + 5.89758i 0 −3.71761 − 6.43909i −15.3830 − 15.3830i 0 −22.7195
325.1 −0.983199 + 3.66935i 0 −9.03333 − 5.21539i −2.06524 − 7.70758i 0 −0.830601 + 1.43864i 17.2741 − 17.2741i 0 30.3123
325.2 −0.540554 + 2.01737i 0 −0.313497 − 0.180998i −0.199493 − 0.744517i 0 4.01523 − 6.95457i −5.37268 + 5.37268i 0 1.60981
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.a 24
3.b odd 2 1 37.3.g.a ✓ 24
37.g odd 12 1 inner 333.3.bb.a 24
111.m even 12 1 37.3.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.3.g.a ✓ 24 3.b odd 2 1
37.3.g.a ✓ 24 111.m even 12 1
333.3.bb.a 24 1.a even 1 1 trivial
333.3.bb.a 24 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 12 T_{2}^{22} + 8 T_{2}^{21} - 123 T_{2}^{20} + 88 T_{2}^{19} - 2020 T_{2}^{18} + \cdots + 3017169 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display