Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [375,3,Mod(374,375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(375, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("375.374");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 375 = 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 375.d (of order \(2\), degree \(1\), 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: | \(10.2180099135\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
374.1 | −3.76125 | −1.36465 | − | 2.67165i | 10.1470 | 0 | 5.13278 | + | 10.0488i | 11.2763i | −23.1204 | −5.27548 | + | 7.29173i | 0 | ||||||||||||
374.2 | −3.76125 | −1.36465 | + | 2.67165i | 10.1470 | 0 | 5.13278 | − | 10.0488i | − | 11.2763i | −23.1204 | −5.27548 | − | 7.29173i | 0 | |||||||||||
374.3 | −3.38842 | −2.11620 | − | 2.12643i | 7.48139 | 0 | 7.17057 | + | 7.20524i | − | 6.70269i | −11.7964 | −0.0434208 | + | 8.99990i | 0 | |||||||||||
374.4 | −3.38842 | −2.11620 | + | 2.12643i | 7.48139 | 0 | 7.17057 | − | 7.20524i | 6.70269i | −11.7964 | −0.0434208 | − | 8.99990i | 0 | ||||||||||||
374.5 | −2.98144 | 2.72527 | − | 1.25416i | 4.88897 | 0 | −8.12522 | + | 3.73919i | 12.5530i | −2.65041 | 5.85419 | − | 6.83583i | 0 | ||||||||||||
374.6 | −2.98144 | 2.72527 | + | 1.25416i | 4.88897 | 0 | −8.12522 | − | 3.73919i | − | 12.5530i | −2.65041 | 5.85419 | + | 6.83583i | 0 | |||||||||||
374.7 | −2.76294 | 2.51808 | − | 1.63072i | 3.63386 | 0 | −6.95733 | + | 4.50559i | − | 1.45760i | 1.01162 | 3.68150 | − | 8.21259i | 0 | |||||||||||
374.8 | −2.76294 | 2.51808 | + | 1.63072i | 3.63386 | 0 | −6.95733 | − | 4.50559i | 1.45760i | 1.01162 | 3.68150 | + | 8.21259i | 0 | ||||||||||||
374.9 | −1.84620 | −2.91022 | − | 0.728424i | −0.591546 | 0 | 5.37285 | + | 1.34482i | − | 11.6440i | 8.47691 | 7.93880 | + | 4.23975i | 0 | |||||||||||
374.10 | −1.84620 | −2.91022 | + | 0.728424i | −0.591546 | 0 | 5.37285 | − | 1.34482i | 11.6440i | 8.47691 | 7.93880 | − | 4.23975i | 0 | ||||||||||||
374.11 | −1.67701 | −0.0488427 | − | 2.99960i | −1.18764 | 0 | 0.0819097 | + | 5.03036i | − | 0.986092i | 8.69972 | −8.99523 | + | 0.293018i | 0 | |||||||||||
374.12 | −1.67701 | −0.0488427 | + | 2.99960i | −1.18764 | 0 | 0.0819097 | − | 5.03036i | 0.986092i | 8.69972 | −8.99523 | − | 0.293018i | 0 | ||||||||||||
374.13 | −0.654766 | 1.42077 | − | 2.64224i | −3.57128 | 0 | −0.930269 | + | 1.73005i | 1.39150i | 4.95742 | −4.96285 | − | 7.50800i | 0 | ||||||||||||
374.14 | −0.654766 | 1.42077 | + | 2.64224i | −3.57128 | 0 | −0.930269 | − | 1.73005i | − | 1.39150i | 4.95742 | −4.96285 | + | 7.50800i | 0 | |||||||||||
374.15 | −0.446371 | −2.81092 | − | 1.04821i | −3.80075 | 0 | 1.25471 | + | 0.467892i | 6.22297i | 3.48203 | 6.80250 | + | 5.89288i | 0 | ||||||||||||
374.16 | −0.446371 | −2.81092 | + | 1.04821i | −3.80075 | 0 | 1.25471 | − | 0.467892i | − | 6.22297i | 3.48203 | 6.80250 | − | 5.89288i | 0 | |||||||||||
374.17 | 0.446371 | 2.81092 | − | 1.04821i | −3.80075 | 0 | 1.25471 | − | 0.467892i | 6.22297i | −3.48203 | 6.80250 | − | 5.89288i | 0 | ||||||||||||
374.18 | 0.446371 | 2.81092 | + | 1.04821i | −3.80075 | 0 | 1.25471 | + | 0.467892i | − | 6.22297i | −3.48203 | 6.80250 | + | 5.89288i | 0 | |||||||||||
374.19 | 0.654766 | −1.42077 | − | 2.64224i | −3.57128 | 0 | −0.930269 | − | 1.73005i | 1.39150i | −4.95742 | −4.96285 | + | 7.50800i | 0 | ||||||||||||
374.20 | 0.654766 | −1.42077 | + | 2.64224i | −3.57128 | 0 | −0.930269 | + | 1.73005i | − | 1.39150i | −4.95742 | −4.96285 | − | 7.50800i | 0 | |||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 375.3.d.e | 32 | |
3.b | odd | 2 | 1 | inner | 375.3.d.e | 32 | |
5.b | even | 2 | 1 | inner | 375.3.d.e | 32 | |
5.c | odd | 4 | 2 | 375.3.c.c | ✓ | 32 | |
15.d | odd | 2 | 1 | inner | 375.3.d.e | 32 | |
15.e | even | 4 | 2 | 375.3.c.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
375.3.c.c | ✓ | 32 | 5.c | odd | 4 | 2 | |
375.3.c.c | ✓ | 32 | 15.e | even | 4 | 2 | |
375.3.d.e | 32 | 1.a | even | 1 | 1 | trivial | |
375.3.d.e | 32 | 3.b | odd | 2 | 1 | inner | |
375.3.d.e | 32 | 5.b | even | 2 | 1 | inner | |
375.3.d.e | 32 | 15.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 49 T_{2}^{14} + 956 T_{2}^{12} - 9479 T_{2}^{10} + 50466 T_{2}^{8} - 139855 T_{2}^{6} + 179160 T_{2}^{4} - 75825 T_{2}^{2} + 9025 \)
acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\).