Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1323,2,Mod(440,1323)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1323.440");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1323 = 3^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1323.o (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: | \(10.5642081874\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 441) |
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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
440.1 | −2.23278 | − | 1.28910i | 0 | 2.32354 | + | 4.02449i | 1.16595 | + | 2.01948i | 0 | 0 | − | 6.82470i | 0 | − | 6.01207i | ||||||||||
440.2 | −2.23278 | − | 1.28910i | 0 | 2.32354 | + | 4.02449i | −1.16595 | − | 2.01948i | 0 | 0 | − | 6.82470i | 0 | 6.01207i | |||||||||||
440.3 | −1.80506 | − | 1.04215i | 0 | 1.17216 | + | 2.03024i | −1.65233 | − | 2.86191i | 0 | 0 | − | 0.717672i | 0 | 6.88790i | |||||||||||
440.4 | −1.80506 | − | 1.04215i | 0 | 1.17216 | + | 2.03024i | 1.65233 | + | 2.86191i | 0 | 0 | − | 0.717672i | 0 | − | 6.88790i | ||||||||||
440.5 | −1.61855 | − | 0.934468i | 0 | 0.746462 | + | 1.29291i | −1.25287 | − | 2.17003i | 0 | 0 | 0.947692i | 0 | 4.68306i | ||||||||||||
440.6 | −1.61855 | − | 0.934468i | 0 | 0.746462 | + | 1.29291i | 1.25287 | + | 2.17003i | 0 | 0 | 0.947692i | 0 | − | 4.68306i | |||||||||||
440.7 | −1.58658 | − | 0.916012i | 0 | 0.678156 | + | 1.17460i | 0.322784 | + | 0.559079i | 0 | 0 | 1.17925i | 0 | − | 1.18270i | |||||||||||
440.8 | −1.58658 | − | 0.916012i | 0 | 0.678156 | + | 1.17460i | −0.322784 | − | 0.559079i | 0 | 0 | 1.17925i | 0 | 1.18270i | ||||||||||||
440.9 | −0.575298 | − | 0.332148i | 0 | −0.779355 | − | 1.34988i | −0.0141520 | − | 0.0245119i | 0 | 0 | 2.36404i | 0 | 0.0188022i | ||||||||||||
440.10 | −0.575298 | − | 0.332148i | 0 | −0.779355 | − | 1.34988i | 0.0141520 | + | 0.0245119i | 0 | 0 | 2.36404i | 0 | − | 0.0188022i | |||||||||||
440.11 | −0.105953 | − | 0.0611722i | 0 | −0.992516 | − | 1.71909i | −0.264715 | − | 0.458500i | 0 | 0 | 0.487547i | 0 | 0.0647728i | ||||||||||||
440.12 | −0.105953 | − | 0.0611722i | 0 | −0.992516 | − | 1.71909i | 0.264715 | + | 0.458500i | 0 | 0 | 0.487547i | 0 | − | 0.0647728i | |||||||||||
440.13 | 0.367369 | + | 0.212101i | 0 | −0.910027 | − | 1.57621i | 1.80381 | + | 3.12430i | 0 | 0 | − | 1.62047i | 0 | 1.53036i | |||||||||||
440.14 | 0.367369 | + | 0.212101i | 0 | −0.910027 | − | 1.57621i | −1.80381 | − | 3.12430i | 0 | 0 | − | 1.62047i | 0 | − | 1.53036i | ||||||||||
440.15 | 0.850109 | + | 0.490811i | 0 | −0.518210 | − | 0.897565i | 0.940599 | + | 1.62916i | 0 | 0 | − | 2.98061i | 0 | 1.84662i | |||||||||||
440.16 | 0.850109 | + | 0.490811i | 0 | −0.518210 | − | 0.897565i | −0.940599 | − | 1.62916i | 0 | 0 | − | 2.98061i | 0 | − | 1.84662i | ||||||||||
440.17 | 1.02035 | + | 0.589100i | 0 | −0.305921 | − | 0.529871i | −2.16601 | − | 3.75164i | 0 | 0 | − | 3.07728i | 0 | − | 5.10399i | ||||||||||
440.18 | 1.02035 | + | 0.589100i | 0 | −0.305921 | − | 0.529871i | 2.16601 | + | 3.75164i | 0 | 0 | − | 3.07728i | 0 | 5.10399i | |||||||||||
440.19 | 1.28562 | + | 0.742253i | 0 | 0.101880 | + | 0.176462i | −0.154215 | − | 0.267109i | 0 | 0 | − | 2.66653i | 0 | − | 0.457868i | ||||||||||
440.20 | 1.28562 | + | 0.742253i | 0 | 0.101880 | + | 0.176462i | 0.154215 | + | 0.267109i | 0 | 0 | − | 2.66653i | 0 | 0.457868i | |||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
63.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1323.2.o.e | 48 | |
3.b | odd | 2 | 1 | 441.2.o.e | ✓ | 48 | |
7.b | odd | 2 | 1 | inner | 1323.2.o.e | 48 | |
7.c | even | 3 | 1 | 1323.2.i.d | 48 | ||
7.c | even | 3 | 1 | 1323.2.s.d | 48 | ||
7.d | odd | 6 | 1 | 1323.2.i.d | 48 | ||
7.d | odd | 6 | 1 | 1323.2.s.d | 48 | ||
9.c | even | 3 | 1 | 441.2.o.e | ✓ | 48 | |
9.d | odd | 6 | 1 | inner | 1323.2.o.e | 48 | |
21.c | even | 2 | 1 | 441.2.o.e | ✓ | 48 | |
21.g | even | 6 | 1 | 441.2.i.d | 48 | ||
21.g | even | 6 | 1 | 441.2.s.d | 48 | ||
21.h | odd | 6 | 1 | 441.2.i.d | 48 | ||
21.h | odd | 6 | 1 | 441.2.s.d | 48 | ||
63.g | even | 3 | 1 | 441.2.i.d | 48 | ||
63.h | even | 3 | 1 | 441.2.s.d | 48 | ||
63.i | even | 6 | 1 | 1323.2.s.d | 48 | ||
63.j | odd | 6 | 1 | 1323.2.s.d | 48 | ||
63.k | odd | 6 | 1 | 441.2.i.d | 48 | ||
63.l | odd | 6 | 1 | 441.2.o.e | ✓ | 48 | |
63.n | odd | 6 | 1 | 1323.2.i.d | 48 | ||
63.o | even | 6 | 1 | inner | 1323.2.o.e | 48 | |
63.s | even | 6 | 1 | 1323.2.i.d | 48 | ||
63.t | odd | 6 | 1 | 441.2.s.d | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.i.d | 48 | 21.g | even | 6 | 1 | ||
441.2.i.d | 48 | 21.h | odd | 6 | 1 | ||
441.2.i.d | 48 | 63.g | even | 3 | 1 | ||
441.2.i.d | 48 | 63.k | odd | 6 | 1 | ||
441.2.o.e | ✓ | 48 | 3.b | odd | 2 | 1 | |
441.2.o.e | ✓ | 48 | 9.c | even | 3 | 1 | |
441.2.o.e | ✓ | 48 | 21.c | even | 2 | 1 | |
441.2.o.e | ✓ | 48 | 63.l | odd | 6 | 1 | |
441.2.s.d | 48 | 21.g | even | 6 | 1 | ||
441.2.s.d | 48 | 21.h | odd | 6 | 1 | ||
441.2.s.d | 48 | 63.h | even | 3 | 1 | ||
441.2.s.d | 48 | 63.t | odd | 6 | 1 | ||
1323.2.i.d | 48 | 7.c | even | 3 | 1 | ||
1323.2.i.d | 48 | 7.d | odd | 6 | 1 | ||
1323.2.i.d | 48 | 63.n | odd | 6 | 1 | ||
1323.2.i.d | 48 | 63.s | even | 6 | 1 | ||
1323.2.o.e | 48 | 1.a | even | 1 | 1 | trivial | |
1323.2.o.e | 48 | 7.b | odd | 2 | 1 | inner | |
1323.2.o.e | 48 | 9.d | odd | 6 | 1 | inner | |
1323.2.o.e | 48 | 63.o | even | 6 | 1 | inner | |
1323.2.s.d | 48 | 7.c | even | 3 | 1 | ||
1323.2.s.d | 48 | 7.d | odd | 6 | 1 | ||
1323.2.s.d | 48 | 63.i | even | 6 | 1 | ||
1323.2.s.d | 48 | 63.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):
\( T_{2}^{24} - 18 T_{2}^{22} + 207 T_{2}^{20} + 12 T_{2}^{19} - 1434 T_{2}^{18} - 108 T_{2}^{17} + 7227 T_{2}^{16} + 540 T_{2}^{15} - 24510 T_{2}^{14} + 61340 T_{2}^{12} - 9612 T_{2}^{11} - 101154 T_{2}^{10} + 34488 T_{2}^{9} + \cdots + 49 \) |
\( T_{5}^{48} + 72 T_{5}^{46} + 3024 T_{5}^{44} + 85168 T_{5}^{42} + 1793667 T_{5}^{40} + 29110188 T_{5}^{38} + 375560984 T_{5}^{36} + 3876266316 T_{5}^{34} + 32327781627 T_{5}^{32} + 216703289260 T_{5}^{30} + \cdots + 2401 \) |