Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [277,2,Mod(276,277)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(277, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("277.276");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 277 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 277.b (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: | \(2.21185613599\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
276.1 | − | 2.68652i | −2.60737 | −5.21740 | 1.33248i | 7.00475i | 1.65648 | 8.64361i | 3.79837 | 3.57973 | |||||||||||||||||
276.2 | − | 2.56713i | 0.633441 | −4.59016 | − | 3.09865i | − | 1.62613i | 0.825668 | 6.64928i | −2.59875 | −7.95463 | |||||||||||||||
276.3 | − | 2.18038i | 1.68863 | −2.75407 | 2.77637i | − | 3.68186i | 5.06490 | 1.64417i | −0.148525 | 6.05355 | ||||||||||||||||
276.4 | − | 2.08361i | 3.11952 | −2.34144 | − | 1.38208i | − | 6.49986i | −1.57376 | 0.711419i | 6.73139 | −2.87971 | |||||||||||||||
276.5 | − | 2.06854i | −0.276722 | −2.27884 | 0.357229i | 0.572410i | −3.43105 | 0.576785i | −2.92342 | 0.738940 | |||||||||||||||||
276.6 | − | 1.48073i | −2.72109 | −0.192568 | − | 2.57972i | 4.02921i | −2.42120 | − | 2.67632i | 4.40434 | −3.81987 | |||||||||||||||
276.7 | − | 1.36148i | −2.29126 | 0.146365 | 1.83057i | 3.11951i | 2.23488 | − | 2.92224i | 2.24986 | 2.49229 | ||||||||||||||||
276.8 | − | 1.02645i | 1.27949 | 0.946398 | − | 1.00561i | − | 1.31333i | −0.808050 | − | 3.02433i | −1.36290 | −1.03221 | ||||||||||||||
276.9 | − | 0.939915i | 2.45709 | 1.11656 | 4.02488i | − | 2.30946i | −3.10332 | − | 2.92930i | 3.03729 | 3.78305 | |||||||||||||||
276.10 | − | 0.873504i | 0.0630325 | 1.23699 | − | 2.12471i | − | 0.0550591i | 3.67656 | − | 2.82753i | −2.99603 | −1.85594 | ||||||||||||||
276.11 | − | 0.268026i | −1.34476 | 1.92816 | 3.33851i | 0.360432i | −0.121112 | − | 1.05285i | −1.19161 | 0.894808 | ||||||||||||||||
276.12 | 0.268026i | −1.34476 | 1.92816 | − | 3.33851i | − | 0.360432i | −0.121112 | 1.05285i | −1.19161 | 0.894808 | ||||||||||||||||
276.13 | 0.873504i | 0.0630325 | 1.23699 | 2.12471i | 0.0550591i | 3.67656 | 2.82753i | −2.99603 | −1.85594 | ||||||||||||||||||
276.14 | 0.939915i | 2.45709 | 1.11656 | − | 4.02488i | 2.30946i | −3.10332 | 2.92930i | 3.03729 | 3.78305 | |||||||||||||||||
276.15 | 1.02645i | 1.27949 | 0.946398 | 1.00561i | 1.31333i | −0.808050 | 3.02433i | −1.36290 | −1.03221 | ||||||||||||||||||
276.16 | 1.36148i | −2.29126 | 0.146365 | − | 1.83057i | − | 3.11951i | 2.23488 | 2.92224i | 2.24986 | 2.49229 | ||||||||||||||||
276.17 | 1.48073i | −2.72109 | −0.192568 | 2.57972i | − | 4.02921i | −2.42120 | 2.67632i | 4.40434 | −3.81987 | |||||||||||||||||
276.18 | 2.06854i | −0.276722 | −2.27884 | − | 0.357229i | − | 0.572410i | −3.43105 | − | 0.576785i | −2.92342 | 0.738940 | |||||||||||||||
276.19 | 2.08361i | 3.11952 | −2.34144 | 1.38208i | 6.49986i | −1.57376 | − | 0.711419i | 6.73139 | −2.87971 | |||||||||||||||||
276.20 | 2.18038i | 1.68863 | −2.75407 | − | 2.77637i | 3.68186i | 5.06490 | − | 1.64417i | −0.148525 | 6.05355 | ||||||||||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
277.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 277.2.b.a | ✓ | 22 |
277.b | even | 2 | 1 | inner | 277.2.b.a | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
277.2.b.a | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
277.2.b.a | ✓ | 22 | 277.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(277, [\chi])\).