↑

Properties

Label 333.3.bu.c
Level $333$
Weight $3$
Character orbit 333.bu
Analytic conductor $9.074$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 111)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 18 q^{4} + 18 q^{5} - 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 18 q^{4} + 18 q^{5} - 66 q^{8} - 72 q^{13} - 42 q^{14} - 6 q^{16} + 24 q^{17} + 108 q^{19} + 354 q^{20} + 18 q^{25} + 30 q^{26} + 48 q^{28} + 156 q^{29} - 60 q^{31} + 192 q^{32} - 90 q^{34} - 24 q^{35} - 294 q^{37} + 120 q^{38} + 612 q^{40} - 300 q^{41} - 60 q^{43} - 174 q^{44} + 234 q^{46} - 66 q^{47} - 144 q^{49} + 252 q^{50} + 912 q^{52} - 234 q^{53} + 234 q^{55} - 312 q^{56} - 1014 q^{58} + 18 q^{59} - 720 q^{61} + 1092 q^{62} + 54 q^{64} + 54 q^{65} - 708 q^{67} + 408 q^{68} - 228 q^{70} - 234 q^{74} + 90 q^{76} + 18 q^{77} + 360 q^{79} - 924 q^{80} + 1134 q^{82} - 438 q^{83} - 756 q^{85} - 396 q^{86} + 684 q^{88} - 1470 q^{89} + 1170 q^{91} - 1602 q^{92} - 1008 q^{94} + 984 q^{95} - 774 q^{97} + 1038 q^{98}+O(q^{100}) \) 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 −3.47320 + 0.303865i 0 8.03152 − 1.41617i 1.03843 + 2.22691i 0 −0.159736 − 0.0581393i −13.9941 + 3.74970i 0 −4.28334 − 7.41896i
19.2 −1.46740 + 0.128381i 0 −1.80244 + 0.317818i 3.79259 + 8.13323i 0 3.21025 + 1.16843i 8.29537 − 2.22274i 0 −6.60941 − 11.4478i
19.3 −0.301717 + 0.0263968i 0 −3.84889 + 0.678664i −1.26974 − 2.72296i 0 −8.18613 − 2.97951i 2.31356 − 0.619916i 0 0.454978 + 0.788045i
19.4 0.0925418 − 0.00809636i 0 −3.93073 + 0.693094i −1.38877 − 2.97822i 0 10.9163 + 3.97321i −0.717066 + 0.192137i 0 −0.152632 − 0.264366i
19.5 2.44499 − 0.213909i 0 1.99298 − 0.351417i −1.61088 − 3.45454i 0 −7.99634 − 2.91043i −4.68515 + 1.25538i 0 −4.67753 − 8.10173i
19.6 3.10167 − 0.271361i 0 5.60750 − 0.988754i 2.11176 + 4.52869i 0 2.70338 + 0.983949i 5.09460 − 1.36510i 0 7.77891 + 13.4735i
55.1 −0.288184 − 3.29396i 0 −6.82791 + 1.20394i 3.93280 − 1.83389i 0 −5.46052 − 1.98747i 2.51026 + 9.36840i 0 −7.17415 − 12.4260i
55.2 −0.229203 − 2.61980i 0 −2.87161 + 0.506343i −7.02528 + 3.27594i 0 5.64517 + 2.05467i −0.737880 − 2.75380i 0 10.1925 + 17.6540i
55.3 −0.118190 − 1.35092i 0 2.12822 − 0.375263i 4.87697 − 2.27417i 0 9.98681 + 3.63490i −2.16240 − 8.07018i 0 −3.64862 − 6.31959i
55.4 0.129581 + 1.48112i 0 1.76230 − 0.310742i 8.53751 − 3.98111i 0 −6.52182 − 2.37375i 2.22783 + 8.31439i 0 7.00280 + 12.1292i
55.5 0.135828 + 1.55252i 0 1.54735 − 0.272840i −3.79373 + 1.76905i 0 3.40196 + 1.23821i 2.24719 + 8.38664i 0 −3.26178 − 5.64957i
55.6 0.320578 + 3.66423i 0 −9.38457 + 1.65475i −2.44289 + 1.13914i 0 −7.53933 − 2.74409i −5.26391 − 19.6452i 0 −4.95720 − 8.58612i
91.1 −2.73513 − 1.91516i 0 2.44502 + 6.71763i 0.340724 + 3.89449i 0 −0.880453 − 0.738788i 2.72112 − 10.1554i 0 6.52663 − 11.3045i
91.2 −2.03681 − 1.42619i 0 0.746495 + 2.05098i −0.851181 − 9.72904i 0 −7.20843 − 6.04859i −1.16958 + 4.36494i 0 −12.1418 + 21.0301i
91.3 −0.918684 − 0.643270i 0 −0.937895 − 2.57685i −0.0665498 − 0.760667i 0 8.60985 + 7.22452i −1.95705 + 7.30380i 0 −0.428176 + 0.741623i
91.4 −0.534108 − 0.373987i 0 −1.22267 − 3.35927i 0.544238 + 6.22067i 0 −4.07953 − 3.42313i −1.27831 + 4.77071i 0 2.03576 − 3.52605i
91.5 1.40581 + 0.984361i 0 −0.360736 − 0.991114i 0.161245 + 1.84303i 0 −7.26578 − 6.09671i 2.24521 − 8.37923i 0 −1.58753 + 2.74969i
91.6 2.02839 + 1.42029i 0 0.729051 + 2.00305i −0.356877 − 4.07912i 0 4.85229 + 4.07156i 1.19744 − 4.46889i 0 5.06967 − 8.78092i
109.1 −0.288184 + 3.29396i 0 −6.82791 − 1.20394i 3.93280 + 1.83389i 0 −5.46052 + 1.98747i 2.51026 − 9.36840i 0 −7.17415 + 12.4260i
109.2 −0.229203 + 2.61980i 0 −2.87161 − 0.506343i −7.02528 − 3.27594i 0 5.64517 − 2.05467i −0.737880 + 2.75380i 0 10.1925 − 17.6540i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

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.c 72
3.b odd 2 1 111.3.r.a ✓ 72
37.i odd 36 1 inner 333.3.bu.c 72
111.q even 36 1 111.3.r.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.r.a ✓ 72 3.b odd 2 1
111.3.r.a ✓ 72 111.q even 36 1
333.3.bu.c 72 1.a even 1 1 trivial
333.3.bu.c 72 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 9 T_{2}^{70} + 38 T_{2}^{69} + 42 T_{2}^{68} - 42 T_{2}^{67} + 128 T_{2}^{66} + \cdots + 10\!\cdots\!09 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display