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: | \(0\) |
| 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: |
\( x^{9} - 4x^{8} - 6x^{7} + 37x^{6} - 3x^{5} - 100x^{4} + 49x^{3} + 64x^{2} - 20x - 1 \)
|
| 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
\( x^{9} - 4x^{8} - 6x^{7} + 37x^{6} - 3x^{5} - 100x^{4} + 49x^{3} + 64x^{2} - 20x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 9\nu^{2} + 8\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 15\nu^{5} - 25\nu^{4} + 28\nu^{3} + 20\nu^{2} - 2\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 18\nu^{5} + 29\nu^{4} - 43\nu^{3} - 19\nu^{2} + 13\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 18\nu^{5} + 29\nu^{4} - 44\nu^{3} - 19\nu^{2} + 17\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 18\nu^{5} + 29\nu^{4} - 44\nu^{3} - 18\nu^{2} + 17\nu - 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( 3\nu^{8} - 5\nu^{7} - 31\nu^{6} + 42\nu^{5} + 99\nu^{4} - 92\nu^{3} - 91\nu^{2} + 20\nu + 6 \)
|
| \(\beta_{8}\) | \(=\) |
\( 3\nu^{8} - 6\nu^{7} - 29\nu^{6} + 51\nu^{5} + 84\nu^{4} - 117\nu^{3} - 62\nu^{2} + 40\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 4\beta_{6} - 7\beta_{5} + \beta_{4} + \beta_{3} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} - \beta_{6} - 11\beta_{5} + 8\beta_{4} + 2\beta_{3} + \beta_{2} + 16\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{8} + 14\beta_{6} - 47\beta_{5} + 12\beta_{4} + 9\beta_{3} + 3\beta_{2} - \beta _1 + 82 \)
|
| \(\nu^{7}\) | \(=\) |
\( 22\beta_{8} + \beta_{7} - 12\beta_{6} - 92\beta_{5} + 56\beta_{4} + 21\beta_{3} + 15\beta_{2} + 62\beta _1 + 102 \)
|
| \(\nu^{8}\) | \(=\) |
\( 79\beta_{8} + 2\beta_{7} + 37\beta_{6} - 315\beta_{5} + 103\beta_{4} + 67\beta_{3} + 42\beta_{2} - 15\beta _1 + 474 \)
|
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 |
|
−2.06801 | 3.40180 | 2.27668 | −1.35083 | −7.03498 | −3.89763 | −0.572186 | 8.57226 | 2.79353 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00926 | −0.921236 | 2.03711 | 1.18304 | 1.85100 | −0.0134632 | −0.0745557 | −2.15132 | −2.37704 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.810813 | −1.61254 | −1.34258 | −2.66277 | 1.30747 | −0.908483 | 2.71021 | −0.399715 | 2.15901 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0440254 | −0.265516 | −1.99806 | 3.90380 | 0.0116894 | 1.04547 | 0.176016 | −2.92950 | −0.171866 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.330914 | 3.30856 | −1.89050 | 2.73075 | 1.09485 | −0.690230 | −1.28742 | 7.94659 | 0.903643 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.66103 | 1.31479 | 0.759008 | 0.624941 | 2.18389 | 3.40756 | −2.06132 | −1.27134 | 1.03804 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.15683 | 2.60085 | 2.65190 | −3.03225 | 5.60958 | −0.850064 | 1.40604 | 3.76441 | −6.54003 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.20965 | −1.86208 | 2.88254 | 1.38838 | −4.11455 | 3.12163 | 1.95011 | 0.467359 | 3.06782 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.57369 | 0.0353761 | 4.62390 | 1.21494 | 0.0910473 | −3.21479 | 6.75311 | −2.99875 | 3.12688 | ||||||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 9 |
| 3.b | odd | 2 | 1 | 2493.2.a.f | 9 | ||
| 4.b | odd | 2 | 1 | 4432.2.a.g | 9 | ||
| 5.b | even | 2 | 1 | 6925.2.a.g | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 277.2.a.d | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 2493.2.a.f | 9 | 3.b | odd | 2 | 1 | ||
| 4432.2.a.g | 9 | 4.b | odd | 2 | 1 | ||
| 6925.2.a.g | 9 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 4T_{2}^{8} - 6T_{2}^{7} + 37T_{2}^{6} - 3T_{2}^{5} - 100T_{2}^{4} + 49T_{2}^{3} + 64T_{2}^{2} - 20T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(277))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 4 T^{8} + \cdots - 1 \)
|
| $3$ |
\( T^{9} - 6 T^{8} + \cdots - 1 \)
|
| $5$ |
\( T^{9} - 4 T^{8} + \cdots + 145 \)
|
| $7$ |
\( T^{9} + 2 T^{8} + \cdots - 1 \)
|
| $11$ |
\( T^{9} - 2 T^{8} + \cdots + 43 \)
|
| $13$ |
\( T^{9} + 2 T^{8} + \cdots + 461 \)
|
| $17$ |
\( T^{9} - 15 T^{8} + \cdots - 311 \)
|
| $19$ |
\( T^{9} - 13 T^{8} + \cdots + 5843 \)
|
| $23$ |
\( T^{9} - 26 T^{8} + \cdots + 5 \)
|
| $29$ |
\( T^{9} + 4 T^{8} + \cdots + 1119991 \)
|
| $31$ |
\( T^{9} - 12 T^{8} + \cdots + 566735 \)
|
| $37$ |
\( T^{9} + 25 T^{8} + \cdots - 210625 \)
|
| $41$ |
\( T^{9} + 9 T^{8} + \cdots - 1452419 \)
|
| $43$ |
\( T^{9} + 4 T^{8} + \cdots + 170279 \)
|
| $47$ |
\( T^{9} - 26 T^{8} + \cdots + 336091 \)
|
| $53$ |
\( T^{9} - 23 T^{8} + \cdots + 110431 \)
|
| $59$ |
\( T^{9} - 19 T^{8} + \cdots + 1020485 \)
|
| $61$ |
\( T^{9} - 2 T^{8} + \cdots - 182447 \)
|
| $67$ |
\( T^{9} - T^{8} + \cdots - 182781505 \)
|
| $71$ |
\( T^{9} + 13 T^{8} + \cdots - 711187 \)
|
| $73$ |
\( T^{9} + 21 T^{8} + \cdots + 1317725 \)
|
| $79$ |
\( T^{9} + 16 T^{8} + \cdots + 11751911 \)
|
| $83$ |
\( T^{9} - 31 T^{8} + \cdots + 71689 \)
|
| $89$ |
\( T^{9} + 12 T^{8} + \cdots + 8401321 \)
|
| $97$ |
\( T^{9} + 23 T^{8} + \cdots - 8035099 \)
|
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