Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [276,2,Mod(91,276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(276, 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("276.91");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 276 = 2^{2} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 276.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: | \(2.20387109579\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91.1 | −1.36293 | − | 0.377393i | 1.00000i | 1.71515 | + | 1.02872i | − | 2.63738i | 0.377393 | − | 1.36293i | −3.76288 | −1.94939 | − | 2.04936i | −1.00000 | −0.995329 | + | 3.59456i | |||||||
91.2 | −1.36293 | − | 0.377393i | 1.00000i | 1.71515 | + | 1.02872i | 2.63738i | 0.377393 | − | 1.36293i | 3.76288 | −1.94939 | − | 2.04936i | −1.00000 | 0.995329 | − | 3.59456i | ||||||||
91.3 | −1.36293 | + | 0.377393i | − | 1.00000i | 1.71515 | − | 1.02872i | − | 2.63738i | 0.377393 | + | 1.36293i | 3.76288 | −1.94939 | + | 2.04936i | −1.00000 | 0.995329 | + | 3.59456i | ||||||
91.4 | −1.36293 | + | 0.377393i | − | 1.00000i | 1.71515 | − | 1.02872i | 2.63738i | 0.377393 | + | 1.36293i | −3.76288 | −1.94939 | + | 2.04936i | −1.00000 | −0.995329 | − | 3.59456i | |||||||
91.5 | −0.588134 | − | 1.28612i | − | 1.00000i | −1.30820 | + | 1.51282i | − | 2.28672i | −1.28612 | + | 0.588134i | −3.41490 | 2.71506 | + | 0.792755i | −1.00000 | −2.94099 | + | 1.34490i | ||||||
91.6 | −0.588134 | − | 1.28612i | − | 1.00000i | −1.30820 | + | 1.51282i | 2.28672i | −1.28612 | + | 0.588134i | 3.41490 | 2.71506 | + | 0.792755i | −1.00000 | 2.94099 | − | 1.34490i | |||||||
91.7 | −0.588134 | + | 1.28612i | 1.00000i | −1.30820 | − | 1.51282i | − | 2.28672i | −1.28612 | − | 0.588134i | 3.41490 | 2.71506 | − | 0.792755i | −1.00000 | 2.94099 | + | 1.34490i | |||||||
91.8 | −0.588134 | + | 1.28612i | 1.00000i | −1.30820 | − | 1.51282i | 2.28672i | −1.28612 | − | 0.588134i | −3.41490 | 2.71506 | − | 0.792755i | −1.00000 | −2.94099 | − | 1.34490i | ||||||||
91.9 | −0.279557 | − | 1.38631i | 1.00000i | −1.84370 | + | 0.775104i | − | 1.27568i | 1.38631 | − | 0.279557i | 2.49131 | 1.58995 | + | 2.33924i | −1.00000 | −1.76848 | + | 0.356624i | |||||||
91.10 | −0.279557 | − | 1.38631i | 1.00000i | −1.84370 | + | 0.775104i | 1.27568i | 1.38631 | − | 0.279557i | −2.49131 | 1.58995 | + | 2.33924i | −1.00000 | 1.76848 | − | 0.356624i | ||||||||
91.11 | −0.279557 | + | 1.38631i | − | 1.00000i | −1.84370 | − | 0.775104i | − | 1.27568i | 1.38631 | + | 0.279557i | −2.49131 | 1.58995 | − | 2.33924i | −1.00000 | 1.76848 | + | 0.356624i | ||||||
91.12 | −0.279557 | + | 1.38631i | − | 1.00000i | −1.84370 | − | 0.775104i | 1.27568i | 1.38631 | + | 0.279557i | 2.49131 | 1.58995 | − | 2.33924i | −1.00000 | −1.76848 | − | 0.356624i | |||||||
91.13 | 0.714279 | − | 1.22058i | 1.00000i | −0.979610 | − | 1.74366i | − | 3.78153i | 1.22058 | + | 0.714279i | −1.02234 | −2.82799 | − | 0.0497743i | −1.00000 | −4.61564 | − | 2.70107i | |||||||
91.14 | 0.714279 | − | 1.22058i | 1.00000i | −0.979610 | − | 1.74366i | 3.78153i | 1.22058 | + | 0.714279i | 1.02234 | −2.82799 | − | 0.0497743i | −1.00000 | 4.61564 | + | 2.70107i | ||||||||
91.15 | 0.714279 | + | 1.22058i | − | 1.00000i | −0.979610 | + | 1.74366i | − | 3.78153i | 1.22058 | − | 0.714279i | 1.02234 | −2.82799 | + | 0.0497743i | −1.00000 | 4.61564 | − | 2.70107i | ||||||
91.16 | 0.714279 | + | 1.22058i | − | 1.00000i | −0.979610 | + | 1.74366i | 3.78153i | 1.22058 | − | 0.714279i | −1.02234 | −2.82799 | + | 0.0497743i | −1.00000 | −4.61564 | + | 2.70107i | |||||||
91.17 | 1.11292 | − | 0.872582i | − | 1.00000i | 0.477201 | − | 1.94224i | − | 0.970352i | −0.872582 | − | 1.11292i | −4.31859 | −1.16367 | − | 2.57796i | −1.00000 | −0.846712 | − | 1.07993i | ||||||
91.18 | 1.11292 | − | 0.872582i | − | 1.00000i | 0.477201 | − | 1.94224i | 0.970352i | −0.872582 | − | 1.11292i | 4.31859 | −1.16367 | − | 2.57796i | −1.00000 | 0.846712 | + | 1.07993i | |||||||
91.19 | 1.11292 | + | 0.872582i | 1.00000i | 0.477201 | + | 1.94224i | − | 0.970352i | −0.872582 | + | 1.11292i | 4.31859 | −1.16367 | + | 2.57796i | −1.00000 | 0.846712 | − | 1.07993i | |||||||
91.20 | 1.11292 | + | 0.872582i | 1.00000i | 0.477201 | + | 1.94224i | 0.970352i | −0.872582 | + | 1.11292i | −4.31859 | −1.16367 | + | 2.57796i | −1.00000 | −0.846712 | + | 1.07993i | ||||||||
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 | 276.2.e.a | ✓ | 24 |
3.b | odd | 2 | 1 | 828.2.e.f | 24 | ||
4.b | odd | 2 | 1 | inner | 276.2.e.a | ✓ | 24 |
8.b | even | 2 | 1 | 4416.2.i.d | 24 | ||
8.d | odd | 2 | 1 | 4416.2.i.d | 24 | ||
12.b | even | 2 | 1 | 828.2.e.f | 24 | ||
23.b | odd | 2 | 1 | inner | 276.2.e.a | ✓ | 24 |
69.c | even | 2 | 1 | 828.2.e.f | 24 | ||
92.b | even | 2 | 1 | inner | 276.2.e.a | ✓ | 24 |
184.e | odd | 2 | 1 | 4416.2.i.d | 24 | ||
184.h | even | 2 | 1 | 4416.2.i.d | 24 | ||
276.h | odd | 2 | 1 | 828.2.e.f | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
276.2.e.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
276.2.e.a | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
276.2.e.a | ✓ | 24 | 23.b | odd | 2 | 1 | inner |
276.2.e.a | ✓ | 24 | 92.b | even | 2 | 1 | inner |
828.2.e.f | 24 | 3.b | odd | 2 | 1 | ||
828.2.e.f | 24 | 12.b | even | 2 | 1 | ||
828.2.e.f | 24 | 69.c | even | 2 | 1 | ||
828.2.e.f | 24 | 276.h | odd | 2 | 1 | ||
4416.2.i.d | 24 | 8.b | even | 2 | 1 | ||
4416.2.i.d | 24 | 8.d | odd | 2 | 1 | ||
4416.2.i.d | 24 | 184.e | odd | 2 | 1 | ||
4416.2.i.d | 24 | 184.h | even | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(276, [\chi])\).