Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1127,2,Mod(1,1127)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1127, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1127.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1127 = 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1127.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: | \(8.99914030780\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 10x^{5} - x^{4} + 29x^{3} + 9x^{2} - 24x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 161) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 10x^{5} - x^{4} + 29x^{3} + 9x^{2} - 24x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + \nu^{2} + 9\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 8\nu^{3} + 15\nu^{2} - 11\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 9\nu^{4} - 7\nu^{3} - 20\nu^{2} + 7\nu + 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 9\nu^{4} - 8\nu^{3} - 21\nu^{2} + 11\nu + 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{6} + 7\beta_{5} + \beta_{3} - 8\beta_{2} + 19\beta _1 - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} - \beta_{5} + 9\beta_{4} + \beta_{3} + 33\beta_{2} - 2\beta _1 + 65 \)
|
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 |
|
−2.23428 | 1.89313 | 2.99199 | 2.70420 | −4.22978 | 0 | −2.21639 | 0.583942 | −6.04193 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.04500 | −0.886215 | 2.18203 | −1.51297 | 1.81231 | 0 | −0.372248 | −2.21462 | 3.09402 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.20307 | −0.840182 | −0.552632 | 2.60628 | 1.01079 | 0 | 3.07098 | −2.29409 | −3.13552 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.509818 | 0.958194 | −1.74009 | −3.43532 | 0.488505 | 0 | −1.90676 | −2.08186 | −1.75139 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.17057 | 3.34828 | −0.629767 | −0.135011 | 3.91940 | 0 | −3.07833 | 8.21099 | −0.158040 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.39057 | −1.69325 | −0.0663018 | 3.04157 | −2.35459 | 0 | −2.87335 | −0.132917 | 4.22953 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.41138 | 2.22004 | 3.81476 | 0.731257 | 5.35336 | 0 | 4.37609 | 1.92856 | 1.76334 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 1127.2.a.n | 7 | |
| 7.b | odd | 2 | 1 | 1127.2.a.k | 7 | ||
| 7.c | even | 3 | 2 | 161.2.e.a | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.2.e.a | ✓ | 14 | 7.c | even | 3 | 2 | |
| 1127.2.a.k | 7 | 7.b | odd | 2 | 1 | ||
| 1127.2.a.n | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1127))\):
|
\( T_{2}^{7} - 10T_{2}^{5} + T_{2}^{4} + 29T_{2}^{3} - 9T_{2}^{2} - 24T_{2} + 11 \)
|
|
\( T_{3}^{7} - 5T_{3}^{6} + 25T_{3}^{4} - 12T_{3}^{3} - 37T_{3}^{2} + 10T_{3} + 17 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 10 T^{5} + \cdots + 11 \)
|
| $3$ |
\( T^{7} - 5 T^{6} + \cdots + 17 \)
|
| $5$ |
\( T^{7} - 4 T^{6} + \cdots + 11 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 4 T^{6} + \cdots - 1487 \)
|
| $13$ |
\( T^{7} - 14 T^{6} + \cdots + 1521 \)
|
| $17$ |
\( T^{7} - 4 T^{6} + \cdots + 323 \)
|
| $19$ |
\( T^{7} - 9 T^{6} + \cdots + 2149 \)
|
| $23$ |
\( (T - 1)^{7} \)
|
| $29$ |
\( T^{7} + 5 T^{6} + \cdots + 3717 \)
|
| $31$ |
\( T^{7} - 10 T^{6} + \cdots + 9 \)
|
| $37$ |
\( T^{7} + 5 T^{6} + \cdots - 39481 \)
|
| $41$ |
\( T^{7} - 23 T^{6} + \cdots - 11957 \)
|
| $43$ |
\( T^{7} + 11 T^{6} + \cdots - 531 \)
|
| $47$ |
\( T^{7} - 4 T^{6} + \cdots + 29813 \)
|
| $53$ |
\( T^{7} - 2 T^{6} + \cdots - 4573 \)
|
| $59$ |
\( T^{7} - 7 T^{6} + \cdots - 2551 \)
|
| $61$ |
\( T^{7} - 12 T^{6} + \cdots + 77969 \)
|
| $67$ |
\( T^{7} - 2 T^{6} + \cdots - 1950239 \)
|
| $71$ |
\( T^{7} + 28 T^{6} + \cdots - 1877 \)
|
| $73$ |
\( T^{7} - 59 T^{6} + \cdots + 19683 \)
|
| $79$ |
\( T^{7} + 4 T^{6} + \cdots - 211 \)
|
| $83$ |
\( T^{7} + 9 T^{6} + \cdots + 598427 \)
|
| $89$ |
\( T^{7} - T^{6} + \cdots + 36871 \)
|
| $97$ |
\( T^{7} - 35 T^{6} + \cdots - 12863891 \)
|
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