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Properties

Label 333.3.bu.d
Level $333$
Weight $3$
Character orbit 333.bu
Analytic conductor $9.074$
Analytic rank $0$
Dimension $84$
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(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: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 111)
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 18 q^{4} + 18 q^{5} - 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 18 q^{4} + 18 q^{5} - 66 q^{8} + 72 q^{13} + 138 q^{14} - 6 q^{16} + 24 q^{17} + 36 q^{19} - 186 q^{20} + 234 q^{25} + 30 q^{26} - 432 q^{28} + 12 q^{29} - 60 q^{31} + 192 q^{32} - 90 q^{34} - 24 q^{35} + 144 q^{37} - 384 q^{38} - 468 q^{40} - 300 q^{41} - 60 q^{43} - 174 q^{44} - 54 q^{46} - 66 q^{47} + 438 q^{49} - 36 q^{50} + 48 q^{52} + 270 q^{53} - 126 q^{55} - 24 q^{56} + 498 q^{58} + 18 q^{59} - 288 q^{61} + 228 q^{62} - 2322 q^{64} + 1350 q^{65} + 588 q^{67} + 408 q^{68} - 228 q^{70} - 342 q^{74} + 594 q^{76} + 270 q^{77} + 360 q^{79} - 924 q^{80} + 558 q^{82} + 138 q^{83} + 2700 q^{85} - 2808 q^{86} + 396 q^{88} + 258 q^{89} - 852 q^{91} + 378 q^{92} - 144 q^{94} - 276 q^{95} - 126 q^{97} + 174 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.43430 + 0.300462i 0 7.76490 − 1.36916i −0.668514 − 1.43363i 0 −12.3380 − 4.49065i −12.9358 + 3.46614i 0 2.72663 + 4.72266i
19.2 −2.81348 + 0.246148i 0 3.91587 − 0.690473i 1.73985 + 3.73112i 0 9.13204 + 3.32379i 0.0647188 − 0.0173413i 0 −5.81345 − 10.0692i
19.3 −1.00069 + 0.0875489i 0 −2.94552 + 0.519374i −2.71679 − 5.82617i 0 1.53952 + 0.560340i 6.78321 − 1.81755i 0 3.22873 + 5.59233i
19.4 0.0317485 − 0.00277764i 0 −3.93823 + 0.694416i 3.07893 + 6.60279i 0 −11.4450 − 4.16564i −0.246240 + 0.0659797i 0 0.116092 + 0.201077i
19.5 1.70374 − 0.149058i 0 −1.05872 + 0.186680i 2.35622 + 5.05293i 0 10.0432 + 3.65541i −8.38385 + 2.24644i 0 4.76756 + 8.25766i
19.6 2.34474 − 0.205138i 0 1.51650 − 0.267399i −2.67652 − 5.73982i 0 −7.40526 − 2.69529i −5.59305 + 1.49865i 0 −7.45320 − 12.9093i
19.7 3.56513 − 0.311908i 0 8.67361 − 1.52939i 1.56021 + 3.34589i 0 −1.68449 − 0.613106i 16.6183 − 4.45285i 0 6.60597 + 11.4419i
55.1 −0.343993 − 3.93186i 0 −11.4020 + 2.01048i 2.81755 − 1.31385i 0 10.6685 + 3.88300i 7.74102 + 28.8899i 0 −6.13509 − 10.6263i
55.2 −0.194903 − 2.22775i 0 −0.985654 + 0.173797i −3.63721 + 1.69606i 0 2.97045 + 1.08115i −1.73586 − 6.47833i 0 4.48730 + 7.77223i
55.3 −0.187717 − 2.14561i 0 −0.629193 + 0.110944i 6.94055 − 3.23643i 0 −6.74170 − 2.45378i −1.87364 − 6.99250i 0 −8.24699 − 14.2842i
55.4 −0.00937998 − 0.107214i 0 3.92782 − 0.692581i −5.44677 + 2.53987i 0 −6.95756 − 2.53234i −0.222517 − 0.830444i 0 0.323399 + 0.560144i
55.5 0.0896626 + 1.02485i 0 2.89696 − 0.510812i 2.41168 − 1.12458i 0 7.61029 + 2.76992i 1.84831 + 6.89797i 0 1.36877 + 2.37077i
55.6 0.276187 + 3.15684i 0 −5.95010 + 1.04916i −7.61644 + 3.55160i 0 3.39665 + 1.23628i −1.67470 − 6.25005i 0 −13.3154 − 23.0629i
55.7 0.320554 + 3.66395i 0 −9.38252 + 1.65439i 8.61601 − 4.01771i 0 1.21139 + 0.440908i −5.26152 − 19.6362i 0 17.4826 + 30.2807i
91.1 −3.17062 − 2.22009i 0 3.75593 + 10.3193i 0.00897865 + 0.102626i 0 −1.74749 − 1.46632i 6.99409 − 26.1023i 0 0.199372 − 0.345322i
91.2 −1.97394 − 1.38217i 0 0.617965 + 1.69784i −0.588003 − 6.72090i 0 7.15239 + 6.00157i −1.36786 + 5.10492i 0 −8.12872 + 14.0794i
91.3 −1.49504 − 1.04684i 0 −0.228812 − 0.628655i 0.756338 + 8.64498i 0 6.82005 + 5.72270i −2.20550 + 8.23105i 0 7.91912 − 13.7163i
91.4 −0.881307 − 0.617098i 0 −0.972188 − 2.67106i 0.0447898 + 0.511950i 0 −9.18304 − 7.70549i −1.90534 + 7.11083i 0 0.276450 − 0.478825i
91.5 0.747352 + 0.523302i 0 −1.08339 − 2.97659i −0.699198 − 7.99187i 0 2.87226 + 2.41011i 1.69251 − 6.31654i 0 3.65961 − 6.33863i
91.6 1.14012 + 0.798319i 0 −0.705525 − 1.93841i 0.207859 + 2.37584i 0 0.458343 + 0.384595i 2.18402 − 8.15086i 0 −1.65970 + 2.87468i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 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.d 84
3.b odd 2 1 111.3.r.b ✓ 84
37.i odd 36 1 inner 333.3.bu.d 84
111.q even 36 1 111.3.r.b ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.r.b ✓ 84 3.b odd 2 1
111.3.r.b ✓ 84 111.q even 36 1
333.3.bu.d 84 1.a even 1 1 trivial
333.3.bu.d 84 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} + 9 T_{2}^{82} + 38 T_{2}^{81} + 42 T_{2}^{80} - 42 T_{2}^{79} + 1964 T_{2}^{78} + \cdots + 40\!\cdots\!04 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display