Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [277,2,Mod(1,277)]
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([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("277.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 277 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 277.a (trivial) |
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: | yes |
| Analytic conductor: | \(2.21185613599\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 7x^{7} + 24x^{6} + 9x^{5} - 51x^{4} + 9x^{3} + 18x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{9} - 3x^{8} - 7x^{7} + 24x^{6} + 9x^{5} - 51x^{4} + 9x^{3} + 18x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{8} + 8\nu^{7} + 11\nu^{6} - 64\nu^{5} + \nu^{4} + 131\nu^{3} - 44\nu^{2} - 32\nu - 3 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{8} - 8\nu^{7} - 11\nu^{6} + 64\nu^{5} - \nu^{4} - 131\nu^{3} + 49\nu^{2} + 32\nu - 7 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{8} + 11\nu^{7} + 27\nu^{6} - 83\nu^{5} - 23\nu^{4} + 152\nu^{3} - 73\nu^{2} - 9\nu + 9 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{8} + 12\nu^{7} + 14\nu^{6} - 96\nu^{5} + 29\nu^{4} + 194\nu^{3} - 141\nu^{2} - 33\nu + 18 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{8} - 3\nu^{7} - 7\nu^{6} + 24\nu^{5} + 9\nu^{4} - 51\nu^{3} + 9\nu^{2} + 18\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{8} - 16\nu^{7} - 17\nu^{6} + 123\nu^{5} - 47\nu^{4} - 232\nu^{3} + 193\nu^{2} + 24\nu - 24 ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7\nu^{8} - 23\nu^{7} - 41\nu^{6} + 179\nu^{5} - \nu^{4} - 351\nu^{3} + 189\nu^{2} + 57\nu - 17 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 6\beta_{3} + 5\beta_{2} + 2\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{8} - 8\beta_{7} - 7\beta_{6} + 2\beta_{4} + 8\beta_{3} + 2\beta_{2} + 27\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 21\beta_{8} - 10\beta_{7} - 10\beta_{6} + 9\beta_{5} + 11\beta_{4} + 35\beta_{3} + 28\beta_{2} + 23\beta _1 + 37 \)
|
| \(\nu^{7}\) | \(=\) |
\( 70\beta_{8} - 55\beta_{7} - 46\beta_{6} + 4\beta_{5} + 23\beta_{4} + 58\beta_{3} + 26\beta_{2} + 157\beta _1 + 15 \)
|
| \(\nu^{8}\) | \(=\) |
\( 174 \beta_{8} - 85 \beta_{7} - 81 \beta_{6} + 66 \beta_{5} + 89 \beta_{4} + 215 \beta_{3} + 172 \beta_{2} + \cdots + 190 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.66794 | 0.202589 | 5.11791 | −0.357591 | −0.540495 | −1.69442 | −8.31840 | −2.95896 | 0.954031 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.50819 | −3.09586 | 4.29104 | −3.29974 | 7.76501 | 3.68831 | −5.74638 | 6.58433 | 8.27638 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79394 | −3.14543 | 1.21823 | 2.68090 | 5.64273 | −3.63787 | 1.40245 | 6.89376 | −4.80938 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.75562 | 1.36607 | 1.08221 | −3.25748 | −2.39831 | 1.83043 | 1.61129 | −1.13384 | 5.71890 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.848927 | −2.06993 | −1.27932 | 1.28208 | 1.75722 | 3.54694 | 2.78391 | 1.28463 | −1.08839 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.655698 | 1.08525 | −1.57006 | −0.422223 | −0.711597 | −3.62937 | 2.34088 | −1.82223 | 0.276851 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.856258 | −0.323485 | −1.26682 | −1.57099 | −0.276987 | −1.81589 | −2.79724 | −2.89536 | −1.34518 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.08927 | −0.790856 | −0.813490 | −3.45951 | −0.861456 | 2.28992 | −3.06465 | −2.37455 | −3.76834 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.28480 | −3.22835 | 3.22030 | −3.59545 | −7.37612 | −2.57806 | 2.78814 | 7.42222 | −8.21487 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(277\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 277.2.a.c | ✓ | 9 |
| 3.b | odd | 2 | 1 | 2493.2.a.g | 9 | ||
| 4.b | odd | 2 | 1 | 4432.2.a.j | 9 | ||
| 5.b | even | 2 | 1 | 6925.2.a.h | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 277.2.a.c | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 2493.2.a.g | 9 | 3.b | odd | 2 | 1 | ||
| 4432.2.a.j | 9 | 4.b | odd | 2 | 1 | ||
| 6925.2.a.h | 9 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 6T_{2}^{8} + 4T_{2}^{7} - 37T_{2}^{6} - 69T_{2}^{5} + 24T_{2}^{4} + 119T_{2}^{3} + 34T_{2}^{2} - 52T_{2} - 25 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(277))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 6 T^{8} + \cdots - 25 \)
|
| $3$ |
\( T^{9} + 10 T^{8} + \cdots - 5 \)
|
| $5$ |
\( T^{9} + 12 T^{8} + \cdots + 109 \)
|
| $7$ |
\( T^{9} + 2 T^{8} + \cdots + 5743 \)
|
| $11$ |
\( T^{9} + 14 T^{8} + \cdots - 9 \)
|
| $13$ |
\( T^{9} + 2 T^{8} + \cdots + 42085 \)
|
| $17$ |
\( T^{9} + 19 T^{8} + \cdots + 16721 \)
|
| $19$ |
\( T^{9} - T^{8} + \cdots - 409 \)
|
| $23$ |
\( T^{9} + 30 T^{8} + \cdots + 60805 \)
|
| $29$ |
\( T^{9} + 8 T^{8} + \cdots + 255735 \)
|
| $31$ |
\( T^{9} - 162 T^{7} + \cdots - 1548881 \)
|
| $37$ |
\( T^{9} - T^{8} + \cdots - 895861 \)
|
| $41$ |
\( T^{9} + 13 T^{8} + \cdots + 205885 \)
|
| $43$ |
\( T^{9} + 14 T^{8} + \cdots - 404825 \)
|
| $47$ |
\( T^{9} + 10 T^{8} + \cdots + 53808043 \)
|
| $53$ |
\( T^{9} + 21 T^{8} + \cdots - 4146665 \)
|
| $59$ |
\( T^{9} + 45 T^{8} + \cdots + 2800361 \)
|
| $61$ |
\( T^{9} - 6 T^{8} + \cdots + 1763065 \)
|
| $67$ |
\( T^{9} - 5 T^{8} + \cdots - 4839061 \)
|
| $71$ |
\( T^{9} + 17 T^{8} + \cdots + 3598121 \)
|
| $73$ |
\( T^{9} - 11 T^{8} + \cdots + 903313 \)
|
| $79$ |
\( T^{9} - 24 T^{8} + \cdots - 11957625 \)
|
| $83$ |
\( T^{9} + 53 T^{8} + \cdots + 242004997 \)
|
| $89$ |
\( T^{9} - 12 T^{8} + \cdots - 13147335 \)
|
| $97$ |
\( T^{9} + 7 T^{8} + \cdots - 11088247 \)
|
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