Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [276,2,Mod(5,276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(276, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 11, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("276.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 276 = 2^{2} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 276.k (of order \(22\), degree \(10\), 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: | \(80\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | 0 | −1.56788 | − | 0.736032i | 0 | 0.185142 | − | 1.28769i | 0 | −4.35485 | + | 1.98879i | 0 | 1.91651 | + | 2.30802i | 0 | ||||||||||
5.2 | 0 | −1.25009 | + | 1.19887i | 0 | −0.395237 | + | 2.74894i | 0 | 0.714786 | − | 0.326432i | 0 | 0.125437 | − | 2.99738i | 0 | ||||||||||
5.3 | 0 | −1.09629 | − | 1.34095i | 0 | −0.333236 | + | 2.31771i | 0 | 2.57171 | − | 1.17446i | 0 | −0.596294 | + | 2.94014i | 0 | ||||||||||
5.4 | 0 | −1.00876 | + | 1.40798i | 0 | 0.395237 | − | 2.74894i | 0 | 0.714786 | − | 0.326432i | 0 | −0.964814 | − | 2.84062i | 0 | ||||||||||
5.5 | 0 | 0.297274 | − | 1.70635i | 0 | 0.377490 | − | 2.62550i | 0 | 1.06836 | − | 0.487902i | 0 | −2.82326 | − | 1.01451i | 0 | ||||||||||
5.6 | 0 | 0.951673 | + | 1.44718i | 0 | −0.185142 | + | 1.28769i | 0 | −4.35485 | + | 1.98879i | 0 | −1.18864 | + | 2.75448i | 0 | ||||||||||
5.7 | 0 | 1.48332 | + | 0.894295i | 0 | 0.333236 | − | 2.31771i | 0 | 2.57171 | − | 1.17446i | 0 | 1.40047 | + | 2.65305i | 0 | ||||||||||
5.8 | 0 | 1.64667 | − | 0.537087i | 0 | −0.377490 | + | 2.62550i | 0 | 1.06836 | − | 0.487902i | 0 | 2.42308 | − | 1.76881i | 0 | ||||||||||
17.1 | 0 | −1.72769 | + | 0.122800i | 0 | 1.67238 | − | 1.07478i | 0 | −0.700880 | − | 0.100771i | 0 | 2.96984 | − | 0.424320i | 0 | ||||||||||
17.2 | 0 | −1.38704 | − | 1.03737i | 0 | −1.67238 | + | 1.07478i | 0 | −0.700880 | − | 0.100771i | 0 | 0.847741 | + | 2.87773i | 0 | ||||||||||
17.3 | 0 | −1.27466 | + | 1.17271i | 0 | −2.95319 | + | 1.89790i | 0 | 0.486747 | + | 0.0699836i | 0 | 0.249516 | − | 2.98961i | 0 | ||||||||||
17.4 | 0 | −0.438299 | − | 1.67568i | 0 | 2.95319 | − | 1.89790i | 0 | 0.486747 | + | 0.0699836i | 0 | −2.61579 | + | 1.46889i | 0 | ||||||||||
17.5 | 0 | 0.604048 | + | 1.62331i | 0 | 0.998768 | − | 0.641869i | 0 | 3.17327 | + | 0.456247i | 0 | −2.27025 | + | 1.96111i | 0 | ||||||||||
17.6 | 0 | 1.38367 | + | 1.04185i | 0 | −2.86447 | + | 1.84088i | 0 | −2.95914 | − | 0.425460i | 0 | 0.829091 | + | 2.88316i | 0 | ||||||||||
17.7 | 0 | 1.38578 | − | 1.03904i | 0 | −0.998768 | + | 0.641869i | 0 | 3.17327 | + | 0.456247i | 0 | 0.840791 | − | 2.87977i | 0 | ||||||||||
17.8 | 0 | 1.72729 | − | 0.128392i | 0 | 2.86447 | − | 1.84088i | 0 | −2.95914 | − | 0.425460i | 0 | 2.96703 | − | 0.443540i | 0 | ||||||||||
53.1 | 0 | −1.73158 | + | 0.0405044i | 0 | −0.551710 | − | 1.20808i | 0 | 0.335628 | + | 1.14304i | 0 | 2.99672 | − | 0.140273i | 0 | ||||||||||
53.2 | 0 | −1.67064 | − | 0.457117i | 0 | 1.12657 | + | 2.46685i | 0 | −1.33973 | − | 4.56271i | 0 | 2.58209 | + | 1.52736i | 0 | ||||||||||
53.3 | 0 | −0.756167 | + | 1.55827i | 0 | 0.551710 | + | 1.20808i | 0 | 0.335628 | + | 1.14304i | 0 | −1.85642 | − | 2.35663i | 0 | ||||||||||
53.4 | 0 | −0.751560 | − | 1.56050i | 0 | 1.12484 | + | 2.46306i | 0 | 1.26778 | + | 4.31765i | 0 | −1.87032 | + | 2.34562i | 0 | ||||||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 276.2.k.a | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 276.2.k.a | ✓ | 80 |
23.d | odd | 22 | 1 | inner | 276.2.k.a | ✓ | 80 |
69.g | even | 22 | 1 | inner | 276.2.k.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
276.2.k.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
276.2.k.a | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
276.2.k.a | ✓ | 80 | 23.d | odd | 22 | 1 | inner |
276.2.k.a | ✓ | 80 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(276, [\chi])\).