Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,3,Mod(5,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
chi = DirichletCharacter(H, H._module([0, 23]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.o (of order \(54\), degree \(18\), 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: | \(8.82836056527\) |
Analytic rank: | \(0\) |
Dimension: | \(324\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{54})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{54}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | 0 | −2.85677 | + | 0.915910i | 0 | −4.55987 | + | 3.39470i | 0 | 9.74231 | − | 6.40762i | 0 | 7.32222 | − | 5.23308i | 0 | ||||||||||
5.2 | 0 | −2.80896 | − | 1.05344i | 0 | −7.15515 | + | 5.32681i | 0 | −7.59481 | + | 4.99518i | 0 | 6.78053 | + | 5.91815i | 0 | ||||||||||
5.3 | 0 | −2.78702 | + | 1.11018i | 0 | 2.99904 | − | 2.23270i | 0 | −3.97531 | + | 2.61460i | 0 | 6.53498 | − | 6.18822i | 0 | ||||||||||
5.4 | 0 | −2.76434 | − | 1.16550i | 0 | −0.148228 | + | 0.110352i | 0 | 5.09912 | − | 3.35374i | 0 | 6.28320 | + | 6.44371i | 0 | ||||||||||
5.5 | 0 | −2.61407 | − | 1.47195i | 0 | 7.35484 | − | 5.47548i | 0 | 2.16180 | − | 1.42184i | 0 | 4.66673 | + | 7.69556i | 0 | ||||||||||
5.6 | 0 | −1.59067 | + | 2.54358i | 0 | −2.52291 | + | 1.87823i | 0 | −4.60759 | + | 3.03046i | 0 | −3.93956 | − | 8.09196i | 0 | ||||||||||
5.7 | 0 | −1.40095 | − | 2.65280i | 0 | 1.05662 | − | 0.786622i | 0 | −3.63828 | + | 2.39294i | 0 | −5.07467 | + | 7.43288i | 0 | ||||||||||
5.8 | 0 | −0.763445 | + | 2.90123i | 0 | 3.55757 | − | 2.64851i | 0 | 5.32637 | − | 3.50321i | 0 | −7.83430 | − | 4.42986i | 0 | ||||||||||
5.9 | 0 | −0.463613 | − | 2.96396i | 0 | −2.73041 | + | 2.03271i | 0 | −1.12154 | + | 0.737649i | 0 | −8.57013 | + | 2.74826i | 0 | ||||||||||
5.10 | 0 | 0.876176 | + | 2.86920i | 0 | 5.98178 | − | 4.45327i | 0 | −8.57475 | + | 5.63970i | 0 | −7.46463 | + | 5.02785i | 0 | ||||||||||
5.11 | 0 | 0.897733 | + | 2.86253i | 0 | −1.94378 | + | 1.44709i | 0 | 9.49800 | − | 6.24693i | 0 | −7.38815 | + | 5.13957i | 0 | ||||||||||
5.12 | 0 | 1.44773 | − | 2.62756i | 0 | 3.57760 | − | 2.66342i | 0 | 6.85076 | − | 4.50582i | 0 | −4.80816 | − | 7.60799i | 0 | ||||||||||
5.13 | 0 | 1.79639 | − | 2.40271i | 0 | 3.71691 | − | 2.76713i | 0 | −11.2502 | + | 7.39935i | 0 | −2.54600 | − | 8.63237i | 0 | ||||||||||
5.14 | 0 | 1.81604 | + | 2.38789i | 0 | −4.15318 | + | 3.09193i | 0 | −3.40769 | + | 2.24128i | 0 | −2.40402 | + | 8.67299i | 0 | ||||||||||
5.15 | 0 | 1.87615 | − | 2.34095i | 0 | −7.80193 | + | 5.80832i | 0 | 5.92166 | − | 3.89473i | 0 | −1.96012 | − | 8.78396i | 0 | ||||||||||
5.16 | 0 | 2.91181 | − | 0.722057i | 0 | −2.43946 | + | 1.81611i | 0 | −7.41916 | + | 4.87966i | 0 | 7.95727 | − | 4.20498i | 0 | ||||||||||
5.17 | 0 | 2.94286 | + | 0.582717i | 0 | 5.13522 | − | 3.82303i | 0 | 3.31229 | − | 2.17853i | 0 | 8.32088 | + | 3.42971i | 0 | ||||||||||
5.18 | 0 | 2.97960 | + | 0.349247i | 0 | −2.56932 | + | 1.91279i | 0 | 3.67698 | − | 2.41839i | 0 | 8.75605 | + | 2.08123i | 0 | ||||||||||
29.1 | 0 | −2.99940 | − | 0.0601358i | 0 | 2.76376 | − | 2.60747i | 0 | −5.64514 | + | 0.659822i | 0 | 8.99277 | + | 0.360742i | 0 | ||||||||||
29.2 | 0 | −2.97121 | + | 0.414613i | 0 | −0.778741 | + | 0.734705i | 0 | 1.20130 | − | 0.140411i | 0 | 8.65619 | − | 2.46381i | 0 | ||||||||||
See next 80 embeddings (of 324 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
81.h | odd | 54 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 324.3.o.a | ✓ | 324 |
81.h | odd | 54 | 1 | inner | 324.3.o.a | ✓ | 324 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
324.3.o.a | ✓ | 324 | 1.a | even | 1 | 1 | trivial |
324.3.o.a | ✓ | 324 | 81.h | odd | 54 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(324, [\chi])\).