↑

Properties

Label 297.3.i.a
Level $297$
Weight $3$
Character orbit 297.i
Analytic conductor $8.093$
Analytic rank $0$
Dimension $40$
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] = [297,3,Mod(89,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.89");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.i (of order \(6\), degree \(2\), not 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: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{4} + 9 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{4} + 9 q^{5} + 2 q^{7} - 10 q^{13} + 18 q^{14} - 80 q^{16} - 28 q^{19} - 54 q^{20} - 144 q^{23} + 97 q^{25} + 32 q^{28} + 216 q^{29} + 47 q^{31} - 36 q^{32} + 12 q^{34} + 86 q^{37} - 90 q^{38} - 60 q^{40} + 216 q^{41} - 64 q^{43} - 216 q^{46} - 18 q^{47} - 162 q^{49} + 522 q^{50} + 62 q^{52} - 66 q^{55} - 72 q^{56} + 90 q^{58} + 387 q^{59} + 110 q^{61} + 196 q^{64} + 306 q^{65} - 109 q^{67} - 882 q^{68} - 120 q^{70} - 196 q^{73} - 54 q^{74} + 44 q^{76} - 34 q^{79} - 300 q^{82} - 792 q^{83} + 96 q^{85} - 720 q^{86} + 52 q^{91} - 414 q^{92} - 66 q^{94} + 144 q^{95} - 103 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 −3.25973 − 1.88201i 0 5.08389 + 8.80555i −0.522604 + 0.301726i 0 −0.371328 + 0.643158i − 23.2156i 0 2.27140
89.2 −2.93520 − 1.69464i 0 3.74358 + 6.48408i 1.37440 − 0.793508i 0 −0.861549 + 1.49225i − 11.8190i 0 −5.37883
89.3 −2.65451 − 1.53258i 0 2.69761 + 4.67240i 3.49120 − 2.01564i 0 6.44897 − 11.1699i − 4.27658i 0 −12.3565
89.4 −2.21504 − 1.27885i 0 1.27093 + 2.20131i −7.78213 + 4.49301i 0 −5.10406 + 8.84049i 3.72950i 0 22.9836
89.5 −2.15679 − 1.24522i 0 1.10117 + 1.90728i 2.63461 − 1.52109i 0 −1.35992 + 2.35545i 4.47699i 0 −7.57642
89.6 −1.91056 − 1.10306i 0 0.433491 + 0.750829i −4.78210 + 2.76094i 0 1.03154 − 1.78668i 6.91182i 0 12.1820
89.7 −1.42123 − 0.820550i 0 −0.653395 − 1.13171i 5.79952 − 3.34836i 0 −5.20503 + 9.01537i 8.70897i 0 −10.9900
89.8 −0.752204 − 0.434285i 0 −1.62279 − 2.81076i −1.42742 + 0.824124i 0 4.43294 − 7.67807i 6.29330i 0 1.43162
89.9 −0.608340 − 0.351225i 0 −1.75328 − 3.03677i −2.21374 + 1.27810i 0 −1.66296 + 2.88033i 5.27299i 0 1.79561
89.10 −0.532727 − 0.307570i 0 −1.81080 − 3.13640i 7.70259 − 4.44709i 0 4.82703 − 8.36067i 4.68836i 0 −5.47117
89.11 −0.0561082 − 0.0323941i 0 −1.99790 − 3.46047i −4.48460 + 2.58918i 0 2.96156 − 5.12956i 0.518033i 0 0.335497
89.12 0.236959 + 0.136808i 0 −1.96257 − 3.39927i 0.628841 − 0.363061i 0 −5.73739 + 9.93745i − 2.16845i 0 0.198679
89.13 0.727526 + 0.420037i 0 −1.64714 − 2.85293i 6.64280 − 3.83522i 0 −1.93327 + 3.34852i − 6.12774i 0 6.44375
89.14 1.53457 + 0.885984i 0 −0.430063 − 0.744891i −6.00016 + 3.46419i 0 0.0871073 − 0.150874i − 8.61199i 0 −12.2769
89.15 1.77228 + 1.02323i 0 0.0939895 + 0.162795i 3.48999 − 2.01495i 0 1.83480 − 3.17796i − 7.80113i 0 8.24699
89.16 2.18778 + 1.26312i 0 1.19092 + 2.06274i 1.30944 − 0.756007i 0 5.78758 − 10.0244i − 4.08783i 0 3.81970
89.17 2.80636 + 1.62025i 0 3.25042 + 5.62990i −2.84029 + 1.63984i 0 −6.44071 + 11.1556i 8.10399i 0 −10.6278
89.18 2.85503 + 1.64835i 0 3.43413 + 5.94809i 4.33357 − 2.50199i 0 −2.34187 + 4.05625i 9.45583i 0 16.4966
89.19 3.15799 + 1.82327i 0 4.64860 + 8.05161i −8.41538 + 4.85862i 0 1.47162 − 2.54892i 19.3164i 0 −35.4342
89.20 3.22394 + 1.86134i 0 4.92920 + 8.53763i 5.56145 − 3.21091i 0 3.13495 − 5.42990i 21.8090i 0 23.9064
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.i.a 40
3.b odd 2 1 99.3.i.a ✓ 40
9.c even 3 1 99.3.i.a ✓ 40
9.c even 3 1 891.3.b.a 40
9.d odd 6 1 inner 297.3.i.a 40
9.d odd 6 1 891.3.b.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.i.a ✓ 40 3.b odd 2 1
99.3.i.a ✓ 40 9.c even 3 1
297.3.i.a 40 1.a even 1 1 trivial
297.3.i.a 40 9.d odd 6 1 inner
891.3.b.a 40 9.c even 3 1
891.3.b.a 40 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(297, [\chi])\).