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Properties

Label 333.3.bu.b
Level $333$
Weight $3$
Character orbit 333.bu
Analytic conductor $9.074$
Analytic rank $0$
Dimension $60$
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: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 37)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{2} + 6 q^{4} - 6 q^{5} - 12 q^{7} + 78 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{2} + 6 q^{4} - 6 q^{5} - 12 q^{7} + 78 q^{8} - 6 q^{10} + 18 q^{11} - 12 q^{13} - 36 q^{14} + 18 q^{16} - 12 q^{17} - 84 q^{19} - 96 q^{20} - 36 q^{22} + 66 q^{23} - 138 q^{25} - 24 q^{26} + 324 q^{28} - 72 q^{29} + 252 q^{31} + 90 q^{32} + 78 q^{34} - 180 q^{35} + 72 q^{37} + 156 q^{38} - 276 q^{40} + 240 q^{41} - 36 q^{43} + 282 q^{44} - 192 q^{46} + 72 q^{47} - 462 q^{49} - 246 q^{50} - 678 q^{52} - 6 q^{53} - 216 q^{55} + 540 q^{56} + 450 q^{58} - 6 q^{59} + 636 q^{61} - 432 q^{62} + 1116 q^{64} - 690 q^{65} + 198 q^{67} - 630 q^{68} + 342 q^{70} - 276 q^{71} + 300 q^{74} - 402 q^{76} + 162 q^{77} - 732 q^{79} - 168 q^{80} - 1056 q^{82} + 432 q^{83} - 990 q^{85} + 1614 q^{86} - 792 q^{88} + 888 q^{89} + 366 q^{91} + 102 q^{92} + 1164 q^{94} - 1062 q^{95} + 942 q^{97} - 1746 q^{98}+O(q^{100}) \) 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 −2.73696 + 0.239453i 0 3.49438 − 0.616153i −3.30071 − 7.07840i 0 2.02744 + 0.737927i 1.19877 − 0.321209i 0 10.7289 + 18.5829i
19.2 −1.87894 + 0.164386i 0 −0.435832 + 0.0768489i 0.809415 + 1.73580i 0 −2.08567 − 0.759122i 8.09367 − 2.16869i 0 −1.80618 − 3.12840i
19.3 1.25921 − 0.110166i 0 −2.36576 + 0.417148i −1.10059 − 2.36023i 0 4.82832 + 1.75736i −7.81682 + 2.09451i 0 −1.64590 − 2.85078i
19.4 1.27989 − 0.111976i 0 −2.31364 + 0.407958i 1.74871 + 3.75013i 0 −2.28808 − 0.832793i −7.87955 + 2.11132i 0 2.65809 + 4.60395i
19.5 3.66472 − 0.320622i 0 9.38815 − 1.65538i −2.92231 − 6.26692i 0 6.77942 + 2.46751i 19.6607 − 5.26807i 0 −12.7188 − 22.0295i
55.1 −0.278623 − 3.18467i 0 −6.12529 + 1.08005i −2.84468 + 1.32649i 0 −4.10444 − 1.49389i 1.83665 + 6.85448i 0 5.01704 + 8.68977i
55.2 −0.0646002 − 0.738384i 0 3.39819 − 0.599193i 3.62515 − 1.69043i 0 1.91755 + 0.697932i −1.42931 − 5.33426i 0 −1.48238 − 2.56755i
55.3 −0.0354511 − 0.405208i 0 3.77629 − 0.665863i −3.32440 + 1.55019i 0 −5.10076 − 1.85652i −0.824792 − 3.07816i 0 0.746004 + 1.29212i
55.4 0.191919 + 2.19365i 0 −0.836017 + 0.147412i 0.802307 − 0.374122i 0 −11.9286 − 4.34164i 1.79589 + 6.70234i 0 0.974669 + 1.68818i
55.5 0.251537 + 2.87508i 0 −4.26359 + 0.751786i 0.706447 − 0.329422i 0 6.42268 + 2.33766i −0.246022 − 0.918168i 0 1.12481 + 1.94823i
91.1 −2.27558 − 1.59338i 0 1.27134 + 3.49298i 0.0377032 + 0.430949i 0 −1.11364 − 0.934457i −0.203367 + 0.758975i 0 0.600870 − 1.04074i
91.2 −0.0565356 − 0.0395867i 0 −1.36645 − 3.75429i −0.351619 − 4.01902i 0 0.611390 + 0.513017i −0.142819 + 0.533007i 0 −0.139221 + 0.241137i
91.3 1.06273 + 0.744130i 0 −0.792419 − 2.17715i 0.573589 + 6.55615i 0 2.95704 + 2.48125i 2.12108 − 7.91597i 0 −4.26906 + 7.39423i
91.4 2.54157 + 1.77962i 0 1.92442 + 5.28730i −0.697777 − 7.97562i 0 −9.44540 − 7.92563i −1.30623 + 4.87493i 0 12.4202 − 21.5124i
91.5 2.86037 + 2.00285i 0 2.80222 + 7.69904i 0.546351 + 6.24482i 0 −0.805501 − 0.675896i −3.78961 + 14.1430i 0 −10.9447 + 18.9568i
109.1 −0.278623 + 3.18467i 0 −6.12529 − 1.08005i −2.84468 − 1.32649i 0 −4.10444 + 1.49389i 1.83665 − 6.85448i 0 5.01704 − 8.68977i
109.2 −0.0646002 + 0.738384i 0 3.39819 + 0.599193i 3.62515 + 1.69043i 0 1.91755 − 0.697932i −1.42931 + 5.33426i 0 −1.48238 + 2.56755i
109.3 −0.0354511 + 0.405208i 0 3.77629 + 0.665863i −3.32440 − 1.55019i 0 −5.10076 + 1.85652i −0.824792 + 3.07816i 0 0.746004 − 1.29212i
109.4 0.191919 − 2.19365i 0 −0.836017 − 0.147412i 0.802307 + 0.374122i 0 −11.9286 + 4.34164i 1.79589 − 6.70234i 0 0.974669 − 1.68818i
109.5 0.251537 − 2.87508i 0 −4.26359 − 0.751786i 0.706447 + 0.329422i 0 6.42268 − 2.33766i −0.246022 + 0.918168i 0 1.12481 − 1.94823i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.5
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.b 60
3.b odd 2 1 37.3.i.a ✓ 60
37.i odd 36 1 inner 333.3.bu.b 60
111.q even 36 1 37.3.i.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.3.i.a ✓ 60 3.b odd 2 1
37.3.i.a ✓ 60 111.q even 36 1
333.3.bu.b 60 1.a even 1 1 trivial
333.3.bu.b 60 37.i odd 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 12 T_{2}^{59} + 69 T_{2}^{58} - 294 T_{2}^{57} + 1152 T_{2}^{56} - 3780 T_{2}^{55} + \cdots + 33774720841 \) acting on \(S_{3}^{\mathrm{new}}(333, [\chi])\). Copy content Toggle raw display