Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,6,Mod(607,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.607");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.k (of order \(2\), degree \(1\), 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: | \(146.270043669\) |
Analytic rank: | \(0\) |
Dimension: | \(34\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
607.1 | 0 | 9.00000 | 0 | −97.4348 | 0 | 158.656i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.2 | 0 | 9.00000 | 0 | −97.4348 | 0 | − | 158.656i | 0 | 81.0000 | 0 | |||||||||||||||||
607.3 | 0 | 9.00000 | 0 | −86.4267 | 0 | 12.7350i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.4 | 0 | 9.00000 | 0 | −86.4267 | 0 | − | 12.7350i | 0 | 81.0000 | 0 | |||||||||||||||||
607.5 | 0 | 9.00000 | 0 | −75.7862 | 0 | − | 122.693i | 0 | 81.0000 | 0 | |||||||||||||||||
607.6 | 0 | 9.00000 | 0 | −75.7862 | 0 | 122.693i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.7 | 0 | 9.00000 | 0 | −63.9110 | 0 | 214.330i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.8 | 0 | 9.00000 | 0 | −63.9110 | 0 | − | 214.330i | 0 | 81.0000 | 0 | |||||||||||||||||
607.9 | 0 | 9.00000 | 0 | −47.0512 | 0 | 145.543i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.10 | 0 | 9.00000 | 0 | −47.0512 | 0 | − | 145.543i | 0 | 81.0000 | 0 | |||||||||||||||||
607.11 | 0 | 9.00000 | 0 | −38.6945 | 0 | 219.678i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.12 | 0 | 9.00000 | 0 | −38.6945 | 0 | − | 219.678i | 0 | 81.0000 | 0 | |||||||||||||||||
607.13 | 0 | 9.00000 | 0 | −35.6115 | 0 | 92.6135i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.14 | 0 | 9.00000 | 0 | −35.6115 | 0 | − | 92.6135i | 0 | 81.0000 | 0 | |||||||||||||||||
607.15 | 0 | 9.00000 | 0 | −14.7152 | 0 | − | 127.969i | 0 | 81.0000 | 0 | |||||||||||||||||
607.16 | 0 | 9.00000 | 0 | −14.7152 | 0 | 127.969i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.17 | 0 | 9.00000 | 0 | −10.8850 | 0 | − | 36.4238i | 0 | 81.0000 | 0 | |||||||||||||||||
607.18 | 0 | 9.00000 | 0 | −10.8850 | 0 | 36.4238i | 0 | 81.0000 | 0 | ||||||||||||||||||
607.19 | 0 | 9.00000 | 0 | 18.1600 | 0 | − | 5.76476i | 0 | 81.0000 | 0 | |||||||||||||||||
607.20 | 0 | 9.00000 | 0 | 18.1600 | 0 | 5.76476i | 0 | 81.0000 | 0 | ||||||||||||||||||
See all 34 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
76.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.6.k.d | yes | 34 |
4.b | odd | 2 | 1 | 912.6.k.c | ✓ | 34 | |
19.b | odd | 2 | 1 | 912.6.k.c | ✓ | 34 | |
76.d | even | 2 | 1 | inner | 912.6.k.d | yes | 34 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.6.k.c | ✓ | 34 | 4.b | odd | 2 | 1 | |
912.6.k.c | ✓ | 34 | 19.b | odd | 2 | 1 | |
912.6.k.d | yes | 34 | 1.a | even | 1 | 1 | trivial |
912.6.k.d | yes | 34 | 76.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(912, [\chi])\):
\( T_{5}^{17} + 11 T_{5}^{16} - 32595 T_{5}^{15} - 436217 T_{5}^{14} + 420187667 T_{5}^{13} + 6388075161 T_{5}^{12} - 2747590673713 T_{5}^{11} - 44895804542675 T_{5}^{10} + \cdots + 13\!\cdots\!40 \) |
\( T_{31}^{17} - 5440 T_{31}^{16} - 226121244 T_{31}^{15} + 942057892560 T_{31}^{14} + \cdots + 19\!\cdots\!20 \) |