Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [608,2,Mod(335,608)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(608, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("608.335");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 608 = 2^{5} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 608.s (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: | \(4.85490444289\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 152) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
335.1 | 0 | −2.07624 | + | 1.19872i | 0 | −0.418341 | + | 0.241529i | 0 | − | 1.56686i | 0 | 1.37385 | − | 2.37958i | 0 | |||||||||||
335.2 | 0 | −2.07624 | + | 1.19872i | 0 | 0.418341 | − | 0.241529i | 0 | 1.56686i | 0 | 1.37385 | − | 2.37958i | 0 | ||||||||||||
335.3 | 0 | −1.27630 | + | 0.736872i | 0 | −2.34524 | + | 1.35403i | 0 | 3.01597i | 0 | −0.414040 | + | 0.717138i | 0 | ||||||||||||
335.4 | 0 | −1.27630 | + | 0.736872i | 0 | 2.34524 | − | 1.35403i | 0 | − | 3.01597i | 0 | −0.414040 | + | 0.717138i | 0 | |||||||||||
335.5 | 0 | −0.705625 | + | 0.407393i | 0 | −3.59735 | + | 2.07693i | 0 | − | 3.24695i | 0 | −1.16806 | + | 2.02314i | 0 | |||||||||||
335.6 | 0 | −0.705625 | + | 0.407393i | 0 | 3.59735 | − | 2.07693i | 0 | 3.24695i | 0 | −1.16806 | + | 2.02314i | 0 | ||||||||||||
335.7 | 0 | −0.179130 | + | 0.103421i | 0 | −0.520463 | + | 0.300489i | 0 | − | 4.27429i | 0 | −1.47861 | + | 2.56102i | 0 | |||||||||||
335.8 | 0 | −0.179130 | + | 0.103421i | 0 | 0.520463 | − | 0.300489i | 0 | 4.27429i | 0 | −1.47861 | + | 2.56102i | 0 | ||||||||||||
335.9 | 0 | 1.05104 | − | 0.606818i | 0 | −2.45005 | + | 1.41453i | 0 | 0.450769i | 0 | −0.763544 | + | 1.32250i | 0 | ||||||||||||
335.10 | 0 | 1.05104 | − | 0.606818i | 0 | 2.45005 | − | 1.41453i | 0 | − | 0.450769i | 0 | −0.763544 | + | 1.32250i | 0 | |||||||||||
335.11 | 0 | 2.03079 | − | 1.17247i | 0 | −1.50560 | + | 0.869259i | 0 | 2.63359i | 0 | 1.24939 | − | 2.16401i | 0 | ||||||||||||
335.12 | 0 | 2.03079 | − | 1.17247i | 0 | 1.50560 | − | 0.869259i | 0 | − | 2.63359i | 0 | 1.24939 | − | 2.16401i | 0 | |||||||||||
335.13 | 0 | 2.65547 | − | 1.53314i | 0 | −2.25688 | + | 1.30301i | 0 | − | 4.30088i | 0 | 3.20101 | − | 5.54431i | 0 | |||||||||||
335.14 | 0 | 2.65547 | − | 1.53314i | 0 | 2.25688 | − | 1.30301i | 0 | 4.30088i | 0 | 3.20101 | − | 5.54431i | 0 | ||||||||||||
559.1 | 0 | −2.07624 | − | 1.19872i | 0 | −0.418341 | − | 0.241529i | 0 | 1.56686i | 0 | 1.37385 | + | 2.37958i | 0 | ||||||||||||
559.2 | 0 | −2.07624 | − | 1.19872i | 0 | 0.418341 | + | 0.241529i | 0 | − | 1.56686i | 0 | 1.37385 | + | 2.37958i | 0 | |||||||||||
559.3 | 0 | −1.27630 | − | 0.736872i | 0 | −2.34524 | − | 1.35403i | 0 | − | 3.01597i | 0 | −0.414040 | − | 0.717138i | 0 | |||||||||||
559.4 | 0 | −1.27630 | − | 0.736872i | 0 | 2.34524 | + | 1.35403i | 0 | 3.01597i | 0 | −0.414040 | − | 0.717138i | 0 | ||||||||||||
559.5 | 0 | −0.705625 | − | 0.407393i | 0 | −3.59735 | − | 2.07693i | 0 | 3.24695i | 0 | −1.16806 | − | 2.02314i | 0 | ||||||||||||
559.6 | 0 | −0.705625 | − | 0.407393i | 0 | 3.59735 | + | 2.07693i | 0 | − | 3.24695i | 0 | −1.16806 | − | 2.02314i | 0 | |||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
152.o | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 608.2.s.c | 28 | |
4.b | odd | 2 | 1 | 152.2.o.c | ✓ | 28 | |
8.b | even | 2 | 1 | 152.2.o.c | ✓ | 28 | |
8.d | odd | 2 | 1 | inner | 608.2.s.c | 28 | |
19.d | odd | 6 | 1 | inner | 608.2.s.c | 28 | |
76.f | even | 6 | 1 | 152.2.o.c | ✓ | 28 | |
152.l | odd | 6 | 1 | 152.2.o.c | ✓ | 28 | |
152.o | even | 6 | 1 | inner | 608.2.s.c | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
152.2.o.c | ✓ | 28 | 4.b | odd | 2 | 1 | |
152.2.o.c | ✓ | 28 | 8.b | even | 2 | 1 | |
152.2.o.c | ✓ | 28 | 76.f | even | 6 | 1 | |
152.2.o.c | ✓ | 28 | 152.l | odd | 6 | 1 | |
608.2.s.c | 28 | 1.a | even | 1 | 1 | trivial | |
608.2.s.c | 28 | 8.d | odd | 2 | 1 | inner | |
608.2.s.c | 28 | 19.d | odd | 6 | 1 | inner | |
608.2.s.c | 28 | 152.o | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{14} - 3 T_{3}^{13} - 8 T_{3}^{12} + 33 T_{3}^{11} + 65 T_{3}^{10} - 210 T_{3}^{9} - 201 T_{3}^{8} + 657 T_{3}^{7} + 741 T_{3}^{6} - 834 T_{3}^{5} - 805 T_{3}^{4} + 609 T_{3}^{3} + 904 T_{3}^{2} + 261 T_{3} + 27 \)
acting on \(S_{2}^{\mathrm{new}}(608, [\chi])\).