Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [828,2,Mod(91,828)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("828.91");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 828.e (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: | \(6.61161328736\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 276) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91.1 | −1.40342 | − | 0.174423i | 0 | 1.93915 | + | 0.489577i | − | 2.63558i | 0 | −0.526424 | −2.63604 | − | 1.02531i | 0 | −0.459706 | + | 3.69881i | |||||||||
91.2 | −1.40342 | − | 0.174423i | 0 | 1.93915 | + | 0.489577i | 2.63558i | 0 | 0.526424 | −2.63604 | − | 1.02531i | 0 | 0.459706 | − | 3.69881i | ||||||||||
91.3 | −1.40342 | + | 0.174423i | 0 | 1.93915 | − | 0.489577i | − | 2.63558i | 0 | 0.526424 | −2.63604 | + | 1.02531i | 0 | 0.459706 | + | 3.69881i | |||||||||
91.4 | −1.40342 | + | 0.174423i | 0 | 1.93915 | − | 0.489577i | 2.63558i | 0 | −0.526424 | −2.63604 | + | 1.02531i | 0 | −0.459706 | − | 3.69881i | ||||||||||
91.5 | −1.11292 | − | 0.872582i | 0 | 0.477201 | + | 1.94224i | − | 0.970352i | 0 | −4.31859 | 1.16367 | − | 2.57796i | 0 | −0.846712 | + | 1.07993i | |||||||||
91.6 | −1.11292 | − | 0.872582i | 0 | 0.477201 | + | 1.94224i | 0.970352i | 0 | 4.31859 | 1.16367 | − | 2.57796i | 0 | 0.846712 | − | 1.07993i | ||||||||||
91.7 | −1.11292 | + | 0.872582i | 0 | 0.477201 | − | 1.94224i | − | 0.970352i | 0 | 4.31859 | 1.16367 | + | 2.57796i | 0 | 0.846712 | + | 1.07993i | |||||||||
91.8 | −1.11292 | + | 0.872582i | 0 | 0.477201 | − | 1.94224i | 0.970352i | 0 | −4.31859 | 1.16367 | + | 2.57796i | 0 | −0.846712 | − | 1.07993i | ||||||||||
91.9 | −0.714279 | − | 1.22058i | 0 | −0.979610 | + | 1.74366i | − | 3.78153i | 0 | −1.02234 | 2.82799 | − | 0.0497743i | 0 | −4.61564 | + | 2.70107i | |||||||||
91.10 | −0.714279 | − | 1.22058i | 0 | −0.979610 | + | 1.74366i | 3.78153i | 0 | 1.02234 | 2.82799 | − | 0.0497743i | 0 | 4.61564 | − | 2.70107i | ||||||||||
91.11 | −0.714279 | + | 1.22058i | 0 | −0.979610 | − | 1.74366i | − | 3.78153i | 0 | 1.02234 | 2.82799 | + | 0.0497743i | 0 | 4.61564 | + | 2.70107i | |||||||||
91.12 | −0.714279 | + | 1.22058i | 0 | −0.979610 | − | 1.74366i | 3.78153i | 0 | −1.02234 | 2.82799 | + | 0.0497743i | 0 | −4.61564 | − | 2.70107i | ||||||||||
91.13 | 0.279557 | − | 1.38631i | 0 | −1.84370 | − | 0.775104i | − | 1.27568i | 0 | 2.49131 | −1.58995 | + | 2.33924i | 0 | −1.76848 | − | 0.356624i | |||||||||
91.14 | 0.279557 | − | 1.38631i | 0 | −1.84370 | − | 0.775104i | 1.27568i | 0 | −2.49131 | −1.58995 | + | 2.33924i | 0 | 1.76848 | + | 0.356624i | ||||||||||
91.15 | 0.279557 | + | 1.38631i | 0 | −1.84370 | + | 0.775104i | − | 1.27568i | 0 | −2.49131 | −1.58995 | − | 2.33924i | 0 | 1.76848 | − | 0.356624i | |||||||||
91.16 | 0.279557 | + | 1.38631i | 0 | −1.84370 | + | 0.775104i | 1.27568i | 0 | 2.49131 | −1.58995 | − | 2.33924i | 0 | −1.76848 | + | 0.356624i | ||||||||||
91.17 | 0.588134 | − | 1.28612i | 0 | −1.30820 | − | 1.51282i | − | 2.28672i | 0 | −3.41490 | −2.71506 | + | 0.792755i | 0 | −2.94099 | − | 1.34490i | |||||||||
91.18 | 0.588134 | − | 1.28612i | 0 | −1.30820 | − | 1.51282i | 2.28672i | 0 | 3.41490 | −2.71506 | + | 0.792755i | 0 | 2.94099 | + | 1.34490i | ||||||||||
91.19 | 0.588134 | + | 1.28612i | 0 | −1.30820 | + | 1.51282i | − | 2.28672i | 0 | 3.41490 | −2.71506 | − | 0.792755i | 0 | 2.94099 | − | 1.34490i | |||||||||
91.20 | 0.588134 | + | 1.28612i | 0 | −1.30820 | + | 1.51282i | 2.28672i | 0 | −3.41490 | −2.71506 | − | 0.792755i | 0 | −2.94099 | + | 1.34490i | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
92.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 828.2.e.f | 24 | |
3.b | odd | 2 | 1 | 276.2.e.a | ✓ | 24 | |
4.b | odd | 2 | 1 | inner | 828.2.e.f | 24 | |
12.b | even | 2 | 1 | 276.2.e.a | ✓ | 24 | |
23.b | odd | 2 | 1 | inner | 828.2.e.f | 24 | |
24.f | even | 2 | 1 | 4416.2.i.d | 24 | ||
24.h | odd | 2 | 1 | 4416.2.i.d | 24 | ||
69.c | even | 2 | 1 | 276.2.e.a | ✓ | 24 | |
92.b | even | 2 | 1 | inner | 828.2.e.f | 24 | |
276.h | odd | 2 | 1 | 276.2.e.a | ✓ | 24 | |
552.b | even | 2 | 1 | 4416.2.i.d | 24 | ||
552.h | odd | 2 | 1 | 4416.2.i.d | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
276.2.e.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
276.2.e.a | ✓ | 24 | 12.b | even | 2 | 1 | |
276.2.e.a | ✓ | 24 | 69.c | even | 2 | 1 | |
276.2.e.a | ✓ | 24 | 276.h | odd | 2 | 1 | |
828.2.e.f | 24 | 1.a | even | 1 | 1 | trivial | |
828.2.e.f | 24 | 4.b | odd | 2 | 1 | inner | |
828.2.e.f | 24 | 23.b | odd | 2 | 1 | inner | |
828.2.e.f | 24 | 92.b | even | 2 | 1 | inner | |
4416.2.i.d | 24 | 24.f | even | 2 | 1 | ||
4416.2.i.d | 24 | 24.h | odd | 2 | 1 | ||
4416.2.i.d | 24 | 552.b | even | 2 | 1 | ||
4416.2.i.d | 24 | 552.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\):
\( T_{5}^{12} + 36T_{5}^{10} + 482T_{5}^{8} + 3048T_{5}^{6} + 9312T_{5}^{4} + 12320T_{5}^{2} + 5536 \) |
\( T_{7}^{12} - 52T_{7}^{10} + 990T_{7}^{8} - 8328T_{7}^{6} + 28760T_{7}^{4} - 27328T_{7}^{2} + 5536 \) |