Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,2,Mod(16,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.k (of order \(9\), 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: | \(1.07798042729\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −1.76334 | − | 1.47962i | 1.47207 | + | 0.912694i | 0.572799 | + | 3.24850i | 0.939693 | + | 0.342020i | −1.24532 | − | 3.78748i | −0.745456 | + | 4.22769i | 1.49463 | − | 2.58877i | 1.33398 | + | 2.68710i | −1.15094 | − | 1.99348i |
16.2 | −0.531925 | − | 0.446338i | 0.942993 | − | 1.45285i | −0.263570 | − | 1.49478i | 0.939693 | + | 0.342020i | −1.15006 | + | 0.351912i | −0.0506352 | + | 0.287167i | −1.22136 | + | 2.11545i | −1.22153 | − | 2.74005i | −0.347189 | − | 0.601349i |
16.3 | 0.186537 | + | 0.156523i | −1.71306 | + | 0.255766i | −0.337000 | − | 1.91122i | 0.939693 | + | 0.342020i | −0.359583 | − | 0.220424i | 0.688903 | − | 3.90696i | 0.479795 | − | 0.831029i | 2.86917 | − | 0.876286i | 0.121753 | + | 0.210883i |
16.4 | 1.25101 | + | 1.04972i | −0.905836 | + | 1.47630i | 0.115814 | + | 0.656812i | 0.939693 | + | 0.342020i | −2.68292 | + | 0.895989i | −0.576430 | + | 3.26909i | 1.08849 | − | 1.88532i | −1.35892 | − | 2.67457i | 0.816539 | + | 1.41429i |
16.5 | 1.62376 | + | 1.36249i | −0.235856 | − | 1.71592i | 0.432902 | + | 2.45511i | 0.939693 | + | 0.342020i | 1.95495 | − | 3.10759i | −0.0109747 | + | 0.0622407i | −0.522478 | + | 0.904959i | −2.88874 | + | 0.809420i | 1.05983 | + | 1.83568i |
31.1 | −2.22676 | − | 0.810474i | −1.72562 | − | 0.149064i | 2.76950 | + | 2.32388i | −0.173648 | + | 0.984808i | 3.72174 | + | 1.73050i | −0.151115 | + | 0.126801i | −1.91389 | − | 3.31495i | 2.95556 | + | 0.514456i | 1.18483 | − | 2.05219i |
31.2 | −1.38905 | − | 0.505573i | 1.28409 | + | 1.16238i | 0.141770 | + | 0.118959i | −0.173648 | + | 0.984808i | −1.19600 | − | 2.26380i | 2.40771 | − | 2.02030i | 1.34141 | + | 2.32340i | 0.297765 | + | 2.98519i | 0.739099 | − | 1.28016i |
31.3 | −0.282715 | − | 0.102900i | −0.864728 | + | 1.50075i | −1.46275 | − | 1.22739i | −0.173648 | + | 0.984808i | 0.398898 | − | 0.335304i | −3.28585 | + | 2.75715i | 0.588102 | + | 1.01862i | −1.50449 | − | 2.59548i | 0.150430 | − | 0.260552i |
31.4 | 1.22047 | + | 0.444217i | 1.42893 | + | 0.978851i | −0.239858 | − | 0.201265i | −0.173648 | + | 0.984808i | 1.30916 | + | 1.82942i | −1.16252 | + | 0.975467i | −1.50214 | − | 2.60178i | 1.08370 | + | 2.79743i | −0.649401 | + | 1.12480i |
31.5 | 1.73836 | + | 0.632710i | 0.550980 | − | 1.64208i | 1.08947 | + | 0.914177i | −0.173648 | + | 0.984808i | 1.99676 | − | 2.50591i | −0.872404 | + | 0.732034i | −0.534435 | − | 0.925669i | −2.39284 | − | 1.80950i | −0.924960 | + | 1.60208i |
61.1 | −2.22676 | + | 0.810474i | −1.72562 | + | 0.149064i | 2.76950 | − | 2.32388i | −0.173648 | − | 0.984808i | 3.72174 | − | 1.73050i | −0.151115 | − | 0.126801i | −1.91389 | + | 3.31495i | 2.95556 | − | 0.514456i | 1.18483 | + | 2.05219i |
61.2 | −1.38905 | + | 0.505573i | 1.28409 | − | 1.16238i | 0.141770 | − | 0.118959i | −0.173648 | − | 0.984808i | −1.19600 | + | 2.26380i | 2.40771 | + | 2.02030i | 1.34141 | − | 2.32340i | 0.297765 | − | 2.98519i | 0.739099 | + | 1.28016i |
61.3 | −0.282715 | + | 0.102900i | −0.864728 | − | 1.50075i | −1.46275 | + | 1.22739i | −0.173648 | − | 0.984808i | 0.398898 | + | 0.335304i | −3.28585 | − | 2.75715i | 0.588102 | − | 1.01862i | −1.50449 | + | 2.59548i | 0.150430 | + | 0.260552i |
61.4 | 1.22047 | − | 0.444217i | 1.42893 | − | 0.978851i | −0.239858 | + | 0.201265i | −0.173648 | − | 0.984808i | 1.30916 | − | 1.82942i | −1.16252 | − | 0.975467i | −1.50214 | + | 2.60178i | 1.08370 | − | 2.79743i | −0.649401 | − | 1.12480i |
61.5 | 1.73836 | − | 0.632710i | 0.550980 | + | 1.64208i | 1.08947 | − | 0.914177i | −0.173648 | − | 0.984808i | 1.99676 | + | 2.50591i | −0.872404 | − | 0.732034i | −0.534435 | + | 0.925669i | −2.39284 | + | 1.80950i | −0.924960 | − | 1.60208i |
76.1 | −1.76334 | + | 1.47962i | 1.47207 | − | 0.912694i | 0.572799 | − | 3.24850i | 0.939693 | − | 0.342020i | −1.24532 | + | 3.78748i | −0.745456 | − | 4.22769i | 1.49463 | + | 2.58877i | 1.33398 | − | 2.68710i | −1.15094 | + | 1.99348i |
76.2 | −0.531925 | + | 0.446338i | 0.942993 | + | 1.45285i | −0.263570 | + | 1.49478i | 0.939693 | − | 0.342020i | −1.15006 | − | 0.351912i | −0.0506352 | − | 0.287167i | −1.22136 | − | 2.11545i | −1.22153 | + | 2.74005i | −0.347189 | + | 0.601349i |
76.3 | 0.186537 | − | 0.156523i | −1.71306 | − | 0.255766i | −0.337000 | + | 1.91122i | 0.939693 | − | 0.342020i | −0.359583 | + | 0.220424i | 0.688903 | + | 3.90696i | 0.479795 | + | 0.831029i | 2.86917 | + | 0.876286i | 0.121753 | − | 0.210883i |
76.4 | 1.25101 | − | 1.04972i | −0.905836 | − | 1.47630i | 0.115814 | − | 0.656812i | 0.939693 | − | 0.342020i | −2.68292 | − | 0.895989i | −0.576430 | − | 3.26909i | 1.08849 | + | 1.88532i | −1.35892 | + | 2.67457i | 0.816539 | − | 1.41429i |
76.5 | 1.62376 | − | 1.36249i | −0.235856 | + | 1.71592i | 0.432902 | − | 2.45511i | 0.939693 | − | 0.342020i | 1.95495 | + | 3.10759i | −0.0109747 | − | 0.0622407i | −0.522478 | − | 0.904959i | −2.88874 | − | 0.809420i | 1.05983 | − | 1.83568i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 135.2.k.a | ✓ | 30 |
3.b | odd | 2 | 1 | 405.2.k.a | 30 | ||
5.b | even | 2 | 1 | 675.2.l.d | 30 | ||
5.c | odd | 4 | 2 | 675.2.u.c | 60 | ||
27.e | even | 9 | 1 | inner | 135.2.k.a | ✓ | 30 |
27.e | even | 9 | 1 | 3645.2.a.h | 15 | ||
27.f | odd | 18 | 1 | 405.2.k.a | 30 | ||
27.f | odd | 18 | 1 | 3645.2.a.g | 15 | ||
135.p | even | 18 | 1 | 675.2.l.d | 30 | ||
135.r | odd | 36 | 2 | 675.2.u.c | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.k.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
135.2.k.a | ✓ | 30 | 27.e | even | 9 | 1 | inner |
405.2.k.a | 30 | 3.b | odd | 2 | 1 | ||
405.2.k.a | 30 | 27.f | odd | 18 | 1 | ||
675.2.l.d | 30 | 5.b | even | 2 | 1 | ||
675.2.l.d | 30 | 135.p | even | 18 | 1 | ||
675.2.u.c | 60 | 5.c | odd | 4 | 2 | ||
675.2.u.c | 60 | 135.r | odd | 36 | 2 | ||
3645.2.a.g | 15 | 27.f | odd | 18 | 1 | ||
3645.2.a.h | 15 | 27.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} + 7 T_{2}^{27} - 3 T_{2}^{26} - 36 T_{2}^{25} + 169 T_{2}^{24} + 93 T_{2}^{23} - 315 T_{2}^{22} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\).