Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,3,Mod(2,27)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 27.f (of order \(18\), degree \(6\), 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: | \(0.735696713773\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −3.46291 | − | 0.610605i | 1.47633 | + | 2.61160i | 7.86014 | + | 2.86086i | 3.16671 | + | 3.77394i | −3.51774 | − | 9.94519i | −3.18911 | + | 1.16074i | −13.2912 | − | 7.67367i | −4.64090 | + | 7.71116i | −8.66165 | − | 15.0024i |
2.2 | −2.31604 | − | 0.408381i | −1.62484 | − | 2.52189i | 1.43851 | + | 0.523575i | −3.71692 | − | 4.42965i | 2.73330 | + | 6.50435i | 4.57693 | − | 1.66587i | 5.02894 | + | 2.90346i | −3.71981 | + | 8.19530i | 6.79956 | + | 11.7772i |
2.3 | 0.115908 | + | 0.0204377i | 2.32380 | − | 1.89736i | −3.74575 | − | 1.36334i | 3.98394 | + | 4.74788i | 0.308125 | − | 0.172426i | −7.49258 | + | 2.72708i | −0.814011 | − | 0.469969i | 1.80008 | − | 8.81815i | 0.364735 | + | 0.631740i |
2.4 | 1.14332 | + | 0.201599i | 1.10490 | + | 2.78912i | −2.49222 | − | 0.907094i | −3.46013 | − | 4.12362i | 0.700975 | + | 3.41162i | 9.89907 | − | 3.60297i | −6.68824 | − | 3.86146i | −6.55839 | + | 6.16340i | −3.12473 | − | 5.41219i |
2.5 | 2.58003 | + | 0.454929i | −2.92175 | − | 0.680712i | 2.69082 | + | 0.979377i | 0.519123 | + | 0.618667i | −7.22853 | − | 3.08544i | −5.56035 | + | 2.02380i | −2.57852 | − | 1.48871i | 8.07326 | + | 3.97774i | 1.05790 | + | 1.83234i |
5.1 | −2.17948 | − | 2.59740i | −2.99958 | − | 0.0504108i | −1.30178 | + | 7.38274i | −1.19547 | − | 3.28452i | 6.40658 | + | 7.90098i | −1.88718 | − | 10.7027i | 10.2675 | − | 5.92795i | 8.99492 | + | 0.302422i | −5.92572 | + | 10.2637i |
5.2 | −1.24712 | − | 1.48626i | 2.80515 | + | 1.06355i | 0.0409354 | − | 0.232156i | −1.47839 | − | 4.06185i | −1.91764 | − | 5.49555i | 1.54363 | + | 8.75434i | −7.11704 | + | 4.10903i | 6.73771 | + | 5.96685i | −4.19322 | + | 7.26288i |
5.3 | −0.374063 | − | 0.445791i | −0.428198 | − | 2.96928i | 0.635786 | − | 3.60572i | 2.62195 | + | 7.20376i | −1.16351 | + | 1.30159i | 0.231638 | + | 1.31369i | −3.86112 | + | 2.22922i | −8.63329 | + | 2.54288i | 2.23059 | − | 3.86350i |
5.4 | 0.837612 | + | 0.998227i | −0.987122 | + | 2.83295i | 0.399729 | − | 2.26698i | 0.149473 | + | 0.410673i | −3.65475 | + | 1.38754i | −1.05651 | − | 5.99176i | 7.11182 | − | 4.10601i | −7.05118 | − | 5.59293i | −0.284745 | + | 0.493192i |
5.5 | 2.13670 | + | 2.54642i | −2.03568 | − | 2.20363i | −1.22417 | + | 6.94260i | −2.35247 | − | 6.46335i | 1.26173 | − | 9.89219i | 1.10811 | + | 6.28443i | −8.77937 | + | 5.06877i | −0.712002 | + | 8.97179i | 11.4319 | − | 19.8006i |
11.1 | −2.17948 | + | 2.59740i | −2.99958 | + | 0.0504108i | −1.30178 | − | 7.38274i | −1.19547 | + | 3.28452i | 6.40658 | − | 7.90098i | −1.88718 | + | 10.7027i | 10.2675 | + | 5.92795i | 8.99492 | − | 0.302422i | −5.92572 | − | 10.2637i |
11.2 | −1.24712 | + | 1.48626i | 2.80515 | − | 1.06355i | 0.0409354 | + | 0.232156i | −1.47839 | + | 4.06185i | −1.91764 | + | 5.49555i | 1.54363 | − | 8.75434i | −7.11704 | − | 4.10903i | 6.73771 | − | 5.96685i | −4.19322 | − | 7.26288i |
11.3 | −0.374063 | + | 0.445791i | −0.428198 | + | 2.96928i | 0.635786 | + | 3.60572i | 2.62195 | − | 7.20376i | −1.16351 | − | 1.30159i | 0.231638 | − | 1.31369i | −3.86112 | − | 2.22922i | −8.63329 | − | 2.54288i | 2.23059 | + | 3.86350i |
11.4 | 0.837612 | − | 0.998227i | −0.987122 | − | 2.83295i | 0.399729 | + | 2.26698i | 0.149473 | − | 0.410673i | −3.65475 | − | 1.38754i | −1.05651 | + | 5.99176i | 7.11182 | + | 4.10601i | −7.05118 | + | 5.59293i | −0.284745 | − | 0.493192i |
11.5 | 2.13670 | − | 2.54642i | −2.03568 | + | 2.20363i | −1.22417 | − | 6.94260i | −2.35247 | + | 6.46335i | 1.26173 | + | 9.89219i | 1.10811 | − | 6.28443i | −8.77937 | − | 5.06877i | −0.712002 | − | 8.97179i | 11.4319 | + | 19.8006i |
14.1 | −3.46291 | + | 0.610605i | 1.47633 | − | 2.61160i | 7.86014 | − | 2.86086i | 3.16671 | − | 3.77394i | −3.51774 | + | 9.94519i | −3.18911 | − | 1.16074i | −13.2912 | + | 7.67367i | −4.64090 | − | 7.71116i | −8.66165 | + | 15.0024i |
14.2 | −2.31604 | + | 0.408381i | −1.62484 | + | 2.52189i | 1.43851 | − | 0.523575i | −3.71692 | + | 4.42965i | 2.73330 | − | 6.50435i | 4.57693 | + | 1.66587i | 5.02894 | − | 2.90346i | −3.71981 | − | 8.19530i | 6.79956 | − | 11.7772i |
14.3 | 0.115908 | − | 0.0204377i | 2.32380 | + | 1.89736i | −3.74575 | + | 1.36334i | 3.98394 | − | 4.74788i | 0.308125 | + | 0.172426i | −7.49258 | − | 2.72708i | −0.814011 | + | 0.469969i | 1.80008 | + | 8.81815i | 0.364735 | − | 0.631740i |
14.4 | 1.14332 | − | 0.201599i | 1.10490 | − | 2.78912i | −2.49222 | + | 0.907094i | −3.46013 | + | 4.12362i | 0.700975 | − | 3.41162i | 9.89907 | + | 3.60297i | −6.68824 | + | 3.86146i | −6.55839 | − | 6.16340i | −3.12473 | + | 5.41219i |
14.5 | 2.58003 | − | 0.454929i | −2.92175 | + | 0.680712i | 2.69082 | − | 0.979377i | 0.519123 | − | 0.618667i | −7.22853 | + | 3.08544i | −5.56035 | − | 2.02380i | −2.57852 | + | 1.48871i | 8.07326 | − | 3.97774i | 1.05790 | − | 1.83234i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 27.3.f.a | ✓ | 30 |
3.b | odd | 2 | 1 | 81.3.f.a | 30 | ||
4.b | odd | 2 | 1 | 432.3.bc.a | 30 | ||
9.c | even | 3 | 1 | 243.3.f.c | 30 | ||
9.c | even | 3 | 1 | 243.3.f.d | 30 | ||
9.d | odd | 6 | 1 | 243.3.f.a | 30 | ||
9.d | odd | 6 | 1 | 243.3.f.b | 30 | ||
27.e | even | 9 | 1 | 81.3.f.a | 30 | ||
27.e | even | 9 | 1 | 243.3.f.a | 30 | ||
27.e | even | 9 | 1 | 243.3.f.b | 30 | ||
27.e | even | 9 | 1 | 729.3.b.a | 30 | ||
27.f | odd | 18 | 1 | inner | 27.3.f.a | ✓ | 30 |
27.f | odd | 18 | 1 | 243.3.f.c | 30 | ||
27.f | odd | 18 | 1 | 243.3.f.d | 30 | ||
27.f | odd | 18 | 1 | 729.3.b.a | 30 | ||
108.l | even | 18 | 1 | 432.3.bc.a | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.3.f.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
27.3.f.a | ✓ | 30 | 27.f | odd | 18 | 1 | inner |
81.3.f.a | 30 | 3.b | odd | 2 | 1 | ||
81.3.f.a | 30 | 27.e | even | 9 | 1 | ||
243.3.f.a | 30 | 9.d | odd | 6 | 1 | ||
243.3.f.a | 30 | 27.e | even | 9 | 1 | ||
243.3.f.b | 30 | 9.d | odd | 6 | 1 | ||
243.3.f.b | 30 | 27.e | even | 9 | 1 | ||
243.3.f.c | 30 | 9.c | even | 3 | 1 | ||
243.3.f.c | 30 | 27.f | odd | 18 | 1 | ||
243.3.f.d | 30 | 9.c | even | 3 | 1 | ||
243.3.f.d | 30 | 27.f | odd | 18 | 1 | ||
432.3.bc.a | 30 | 4.b | odd | 2 | 1 | ||
432.3.bc.a | 30 | 108.l | even | 18 | 1 | ||
729.3.b.a | 30 | 27.e | even | 9 | 1 | ||
729.3.b.a | 30 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(27, [\chi])\).