Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,2,Mod(724,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("975.724");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 975.bb (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: | \(7.78541419707\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
724.1 | −2.43531 | − | 1.40603i | 0.866025 | + | 0.500000i | 2.95384 | + | 5.11620i | 0 | −1.40603 | − | 2.43531i | 0.577759 | − | 0.333570i | − | 10.9886i | 0.500000 | + | 0.866025i | 0 | |||||
724.2 | −2.12585 | − | 1.22736i | −0.866025 | − | 0.500000i | 2.01283 | + | 3.48633i | 0 | 1.22736 | + | 2.12585i | 3.62470 | − | 2.09272i | − | 4.97244i | 0.500000 | + | 0.866025i | 0 | |||||
724.3 | −1.54501 | − | 0.892010i | −0.866025 | − | 0.500000i | 0.591364 | + | 1.02427i | 0 | 0.892010 | + | 1.54501i | −2.03615 | + | 1.17557i | 1.45803i | 0.500000 | + | 0.866025i | 0 | ||||||
724.4 | −1.40830 | − | 0.813080i | 0.866025 | + | 0.500000i | 0.322200 | + | 0.558066i | 0 | −0.813080 | − | 1.40830i | 2.67150 | − | 1.54239i | 2.20443i | 0.500000 | + | 0.866025i | 0 | ||||||
724.5 | −0.373368 | − | 0.215564i | −0.866025 | − | 0.500000i | −0.907064 | − | 1.57108i | 0 | 0.215564 | + | 0.373368i | 3.53303 | − | 2.03980i | 1.64438i | 0.500000 | + | 0.866025i | 0 | ||||||
724.6 | −0.200615 | − | 0.115825i | 0.866025 | + | 0.500000i | −0.973169 | − | 1.68558i | 0 | −0.115825 | − | 0.200615i | 1.00629 | − | 0.580982i | 0.914171i | 0.500000 | + | 0.866025i | 0 | ||||||
724.7 | 0.200615 | + | 0.115825i | −0.866025 | − | 0.500000i | −0.973169 | − | 1.68558i | 0 | −0.115825 | − | 0.200615i | −1.00629 | + | 0.580982i | − | 0.914171i | 0.500000 | + | 0.866025i | 0 | |||||
724.8 | 0.373368 | + | 0.215564i | 0.866025 | + | 0.500000i | −0.907064 | − | 1.57108i | 0 | 0.215564 | + | 0.373368i | −3.53303 | + | 2.03980i | − | 1.64438i | 0.500000 | + | 0.866025i | 0 | |||||
724.9 | 1.40830 | + | 0.813080i | −0.866025 | − | 0.500000i | 0.322200 | + | 0.558066i | 0 | −0.813080 | − | 1.40830i | −2.67150 | + | 1.54239i | − | 2.20443i | 0.500000 | + | 0.866025i | 0 | |||||
724.10 | 1.54501 | + | 0.892010i | 0.866025 | + | 0.500000i | 0.591364 | + | 1.02427i | 0 | 0.892010 | + | 1.54501i | 2.03615 | − | 1.17557i | − | 1.45803i | 0.500000 | + | 0.866025i | 0 | |||||
724.11 | 2.12585 | + | 1.22736i | 0.866025 | + | 0.500000i | 2.01283 | + | 3.48633i | 0 | 1.22736 | + | 2.12585i | −3.62470 | + | 2.09272i | 4.97244i | 0.500000 | + | 0.866025i | 0 | ||||||
724.12 | 2.43531 | + | 1.40603i | −0.866025 | − | 0.500000i | 2.95384 | + | 5.11620i | 0 | −1.40603 | − | 2.43531i | −0.577759 | + | 0.333570i | 10.9886i | 0.500000 | + | 0.866025i | 0 | ||||||
874.1 | −2.43531 | + | 1.40603i | 0.866025 | − | 0.500000i | 2.95384 | − | 5.11620i | 0 | −1.40603 | + | 2.43531i | 0.577759 | + | 0.333570i | 10.9886i | 0.500000 | − | 0.866025i | 0 | ||||||
874.2 | −2.12585 | + | 1.22736i | −0.866025 | + | 0.500000i | 2.01283 | − | 3.48633i | 0 | 1.22736 | − | 2.12585i | 3.62470 | + | 2.09272i | 4.97244i | 0.500000 | − | 0.866025i | 0 | ||||||
874.3 | −1.54501 | + | 0.892010i | −0.866025 | + | 0.500000i | 0.591364 | − | 1.02427i | 0 | 0.892010 | − | 1.54501i | −2.03615 | − | 1.17557i | − | 1.45803i | 0.500000 | − | 0.866025i | 0 | |||||
874.4 | −1.40830 | + | 0.813080i | 0.866025 | − | 0.500000i | 0.322200 | − | 0.558066i | 0 | −0.813080 | + | 1.40830i | 2.67150 | + | 1.54239i | − | 2.20443i | 0.500000 | − | 0.866025i | 0 | |||||
874.5 | −0.373368 | + | 0.215564i | −0.866025 | + | 0.500000i | −0.907064 | + | 1.57108i | 0 | 0.215564 | − | 0.373368i | 3.53303 | + | 2.03980i | − | 1.64438i | 0.500000 | − | 0.866025i | 0 | |||||
874.6 | −0.200615 | + | 0.115825i | 0.866025 | − | 0.500000i | −0.973169 | + | 1.68558i | 0 | −0.115825 | + | 0.200615i | 1.00629 | + | 0.580982i | − | 0.914171i | 0.500000 | − | 0.866025i | 0 | |||||
874.7 | 0.200615 | − | 0.115825i | −0.866025 | + | 0.500000i | −0.973169 | + | 1.68558i | 0 | −0.115825 | + | 0.200615i | −1.00629 | − | 0.580982i | 0.914171i | 0.500000 | − | 0.866025i | 0 | ||||||
874.8 | 0.373368 | − | 0.215564i | 0.866025 | − | 0.500000i | −0.907064 | + | 1.57108i | 0 | 0.215564 | − | 0.373368i | −3.53303 | − | 2.03980i | 1.64438i | 0.500000 | − | 0.866025i | 0 | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
65.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 975.2.bb.l | 24 | |
5.b | even | 2 | 1 | inner | 975.2.bb.l | 24 | |
5.c | odd | 4 | 1 | 975.2.i.n | ✓ | 12 | |
5.c | odd | 4 | 1 | 975.2.i.p | yes | 12 | |
13.c | even | 3 | 1 | inner | 975.2.bb.l | 24 | |
65.n | even | 6 | 1 | inner | 975.2.bb.l | 24 | |
65.q | odd | 12 | 1 | 975.2.i.n | ✓ | 12 | |
65.q | odd | 12 | 1 | 975.2.i.p | yes | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
975.2.i.n | ✓ | 12 | 5.c | odd | 4 | 1 | |
975.2.i.n | ✓ | 12 | 65.q | odd | 12 | 1 | |
975.2.i.p | yes | 12 | 5.c | odd | 4 | 1 | |
975.2.i.p | yes | 12 | 65.q | odd | 12 | 1 | |
975.2.bb.l | 24 | 1.a | even | 1 | 1 | trivial | |
975.2.bb.l | 24 | 5.b | even | 2 | 1 | inner | |
975.2.bb.l | 24 | 13.c | even | 3 | 1 | inner | |
975.2.bb.l | 24 | 65.n | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):
\( T_{2}^{24} - 20 T_{2}^{22} + 258 T_{2}^{20} - 1984 T_{2}^{18} + 11107 T_{2}^{16} - 40996 T_{2}^{14} + 110618 T_{2}^{12} - 184396 T_{2}^{10} + 203641 T_{2}^{8} - 46276 T_{2}^{6} + 8012 T_{2}^{4} - 400 T_{2}^{2} + \cdots + 16 \) |
\( T_{7}^{24} - 51 T_{7}^{22} + 1654 T_{7}^{20} - 32791 T_{7}^{18} + 474454 T_{7}^{16} - 4624235 T_{7}^{14} + 33010249 T_{7}^{12} - 150245160 T_{7}^{10} + 475417008 T_{7}^{8} - 699292800 T_{7}^{6} + \cdots + 84934656 \) |