Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [891,2,Mod(82,891)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("891.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 891 = 3^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 891.f (of order \(5\), degree \(4\), 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: | \(7.11467082010\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{5})\) |
Twist minimal: | no (minimal twist has level 99) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 | −0.803965 | − | 2.47435i | 0 | −3.85802 | + | 2.80301i | 0.634128 | − | 1.95164i | 0 | −2.58046 | + | 1.87481i | 5.82773 | + | 4.23409i | 0 | −5.33887 | ||||||||
82.2 | −0.585487 | − | 1.80194i | 0 | −1.28618 | + | 0.934461i | 1.08558 | − | 3.34108i | 0 | 2.77655 | − | 2.01728i | −0.628765 | − | 0.456825i | 0 | −6.65603 | ||||||||
82.3 | −0.458430 | − | 1.41090i | 0 | −0.162452 | + | 0.118028i | −0.0223503 | + | 0.0687873i | 0 | −1.57776 | + | 1.14631i | −2.15937 | − | 1.56887i | 0 | 0.107298 | ||||||||
82.4 | −0.450357 | − | 1.38606i | 0 | −0.100295 | + | 0.0728686i | −0.928914 | + | 2.85890i | 0 | 1.33502 | − | 0.969952i | −2.21193 | − | 1.60706i | 0 | 4.38094 | ||||||||
82.5 | 0.0881157 | + | 0.271192i | 0 | 1.55225 | − | 1.12778i | 0.837045 | − | 2.57616i | 0 | 3.29540 | − | 2.39425i | 0.904002 | + | 0.656796i | 0 | 0.772391 | ||||||||
82.6 | 0.209411 | + | 0.644501i | 0 | 1.24651 | − | 0.905639i | 0.0898605 | − | 0.276562i | 0 | −2.71465 | + | 1.97231i | 1.94121 | + | 1.41037i | 0 | 0.197063 | ||||||||
82.7 | 0.311049 | + | 0.957310i | 0 | 0.798344 | − | 0.580031i | −1.23333 | + | 3.79579i | 0 | −2.22988 | + | 1.62010i | 2.43227 | + | 1.76714i | 0 | −4.01737 | ||||||||
82.8 | 0.660945 | + | 2.03418i | 0 | −2.08301 | + | 1.51339i | 0.517908 | − | 1.59396i | 0 | 1.04819 | − | 0.761555i | −0.994512 | − | 0.722555i | 0 | 3.58471 | ||||||||
82.9 | 0.719701 | + | 2.21501i | 0 | −2.77028 | + | 2.01272i | −0.0979645 | + | 0.301504i | 0 | 1.14758 | − | 0.833769i | −2.68358 | − | 1.94973i | 0 | −0.738340 | ||||||||
163.1 | −0.803965 | + | 2.47435i | 0 | −3.85802 | − | 2.80301i | 0.634128 | + | 1.95164i | 0 | −2.58046 | − | 1.87481i | 5.82773 | − | 4.23409i | 0 | −5.33887 | ||||||||
163.2 | −0.585487 | + | 1.80194i | 0 | −1.28618 | − | 0.934461i | 1.08558 | + | 3.34108i | 0 | 2.77655 | + | 2.01728i | −0.628765 | + | 0.456825i | 0 | −6.65603 | ||||||||
163.3 | −0.458430 | + | 1.41090i | 0 | −0.162452 | − | 0.118028i | −0.0223503 | − | 0.0687873i | 0 | −1.57776 | − | 1.14631i | −2.15937 | + | 1.56887i | 0 | 0.107298 | ||||||||
163.4 | −0.450357 | + | 1.38606i | 0 | −0.100295 | − | 0.0728686i | −0.928914 | − | 2.85890i | 0 | 1.33502 | + | 0.969952i | −2.21193 | + | 1.60706i | 0 | 4.38094 | ||||||||
163.5 | 0.0881157 | − | 0.271192i | 0 | 1.55225 | + | 1.12778i | 0.837045 | + | 2.57616i | 0 | 3.29540 | + | 2.39425i | 0.904002 | − | 0.656796i | 0 | 0.772391 | ||||||||
163.6 | 0.209411 | − | 0.644501i | 0 | 1.24651 | + | 0.905639i | 0.0898605 | + | 0.276562i | 0 | −2.71465 | − | 1.97231i | 1.94121 | − | 1.41037i | 0 | 0.197063 | ||||||||
163.7 | 0.311049 | − | 0.957310i | 0 | 0.798344 | + | 0.580031i | −1.23333 | − | 3.79579i | 0 | −2.22988 | − | 1.62010i | 2.43227 | − | 1.76714i | 0 | −4.01737 | ||||||||
163.8 | 0.660945 | − | 2.03418i | 0 | −2.08301 | − | 1.51339i | 0.517908 | + | 1.59396i | 0 | 1.04819 | + | 0.761555i | −0.994512 | + | 0.722555i | 0 | 3.58471 | ||||||||
163.9 | 0.719701 | − | 2.21501i | 0 | −2.77028 | − | 2.01272i | −0.0979645 | − | 0.301504i | 0 | 1.14758 | + | 0.833769i | −2.68358 | + | 1.94973i | 0 | −0.738340 | ||||||||
487.1 | −1.79211 | + | 1.30204i | 0 | 0.898306 | − | 2.76470i | 0.503591 | + | 0.365880i | 0 | 0.797026 | − | 2.45299i | 0.620850 | + | 1.91078i | 0 | −1.37888 | ||||||||
487.2 | −1.25085 | + | 0.908796i | 0 | 0.120683 | − | 0.371424i | −0.478964 | − | 0.347988i | 0 | 0.223095 | − | 0.686616i | −0.768973 | − | 2.36665i | 0 | 0.915362 | ||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 891.2.f.f | 36 | |
3.b | odd | 2 | 1 | 891.2.f.e | 36 | ||
9.c | even | 3 | 2 | 99.2.m.b | ✓ | 72 | |
9.d | odd | 6 | 2 | 297.2.n.b | 72 | ||
11.c | even | 5 | 1 | inner | 891.2.f.f | 36 | |
11.c | even | 5 | 1 | 9801.2.a.cm | 18 | ||
11.d | odd | 10 | 1 | 9801.2.a.co | 18 | ||
33.f | even | 10 | 1 | 9801.2.a.cn | 18 | ||
33.h | odd | 10 | 1 | 891.2.f.e | 36 | ||
33.h | odd | 10 | 1 | 9801.2.a.cp | 18 | ||
99.m | even | 15 | 2 | 99.2.m.b | ✓ | 72 | |
99.m | even | 15 | 2 | 1089.2.e.p | 36 | ||
99.n | odd | 30 | 2 | 297.2.n.b | 72 | ||
99.o | odd | 30 | 2 | 1089.2.e.o | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.2.m.b | ✓ | 72 | 9.c | even | 3 | 2 | |
99.2.m.b | ✓ | 72 | 99.m | even | 15 | 2 | |
297.2.n.b | 72 | 9.d | odd | 6 | 2 | ||
297.2.n.b | 72 | 99.n | odd | 30 | 2 | ||
891.2.f.e | 36 | 3.b | odd | 2 | 1 | ||
891.2.f.e | 36 | 33.h | odd | 10 | 1 | ||
891.2.f.f | 36 | 1.a | even | 1 | 1 | trivial | |
891.2.f.f | 36 | 11.c | even | 5 | 1 | inner | |
1089.2.e.o | 36 | 99.o | odd | 30 | 2 | ||
1089.2.e.p | 36 | 99.m | even | 15 | 2 | ||
9801.2.a.cm | 18 | 11.c | even | 5 | 1 | ||
9801.2.a.cn | 18 | 33.f | even | 10 | 1 | ||
9801.2.a.co | 18 | 11.d | odd | 10 | 1 | ||
9801.2.a.cp | 18 | 33.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - T_{2}^{35} + 15 T_{2}^{34} - 17 T_{2}^{33} + 132 T_{2}^{32} - 121 T_{2}^{31} + 867 T_{2}^{30} + \cdots + 3025 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\).