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: | \(1\) |
| 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: |
\( x^{7} - 10x^{5} - x^{4} + 25x^{3} + x^{2} - 8x + 1 \)
|
| 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
\( x^{7} - 10x^{5} - x^{4} + 25x^{3} + x^{2} - 8x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 9\nu^{2} + \nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 10\nu^{2} + 5\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 3\nu^{3} + 10\nu^{2} + 4\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 10\nu^{3} + 6\nu^{2} - 5\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 8\nu^{4} - 4\nu^{3} - 15\nu^{2} - \nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{6} + 6\beta_{5} + \beta_{4} + 5\beta_{3} + \beta_{2} + 7\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{6} + 8\beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + 29\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 46\beta_{6} + 37\beta_{5} + 10\beta_{4} + 26\beta_{3} + 12\beta_{2} + 49\beta _1 + 72 \)
|
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 |
|
−2.61328 | −3.04376 | 4.82921 | 1.44516 | 7.95419 | 0 | −7.39352 | 6.26448 | −3.77661 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.76233 | 3.03433 | 1.10579 | −1.94522 | −5.34747 | 0 | 1.57589 | 6.20713 | 3.42812 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.499048 | 1.22023 | −1.75095 | 0.0259569 | −0.608953 | 0 | 1.87191 | −1.51104 | −0.0129537 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.134836 | −2.58533 | −1.98182 | −1.71441 | 0.348596 | 0 | 0.536894 | 3.68392 | 0.231165 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.692701 | 0.603330 | −1.52016 | 2.66776 | 0.417927 | 0 | −2.43842 | −2.63599 | 1.84796 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.14090 | −2.43874 | 2.58346 | −0.818530 | −5.22109 | 0 | 1.24912 | 2.94743 | −1.75239 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.17588 | 0.209939 | 2.73447 | −3.66071 | 0.456804 | 0 | 1.59813 | −2.95593 | −7.96529 | ||||||||||||||||||||||||||||||||||||||||
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.l | 7 | |
| 7.b | odd | 2 | 1 | 1127.2.a.m | 7 | ||
| 7.c | even | 3 | 2 | 161.2.e.b | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.2.e.b | ✓ | 14 | 7.c | even | 3 | 2 | |
| 1127.2.a.l | 7 | 1.a | even | 1 | 1 | trivial | |
| 1127.2.a.m | 7 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1127))\):
|
\( T_{2}^{7} - 10T_{2}^{5} + T_{2}^{4} + 25T_{2}^{3} - T_{2}^{2} - 8T_{2} - 1 \)
|
|
\( T_{3}^{7} + 3T_{3}^{6} - 12T_{3}^{5} - 35T_{3}^{4} + 32T_{3}^{3} + 67T_{3}^{2} - 58T_{3} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 10 T^{5} + \cdots - 1 \)
|
| $3$ |
\( T^{7} + 3 T^{6} + \cdots + 9 \)
|
| $5$ |
\( T^{7} + 4 T^{6} + \cdots - 1 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 25 T^{5} + \cdots + 1 \)
|
| $13$ |
\( T^{7} + 18 T^{6} + \cdots - 183 \)
|
| $17$ |
\( T^{7} + 14 T^{6} + \cdots - 369 \)
|
| $19$ |
\( T^{7} + 11 T^{6} + \cdots - 7 \)
|
| $23$ |
\( (T + 1)^{7} \)
|
| $29$ |
\( T^{7} - 3 T^{6} + \cdots - 73159 \)
|
| $31$ |
\( T^{7} + 6 T^{6} + \cdots - 34227 \)
|
| $37$ |
\( T^{7} + T^{6} + \cdots - 46273 \)
|
| $41$ |
\( T^{7} + 9 T^{6} + \cdots - 19917 \)
|
| $43$ |
\( T^{7} - 9 T^{6} + \cdots - 78303 \)
|
| $47$ |
\( T^{7} - 12 T^{6} + \cdots - 28131 \)
|
| $53$ |
\( T^{7} + 12 T^{6} + \cdots + 223 \)
|
| $59$ |
\( T^{7} - 7 T^{6} + \cdots - 6147 \)
|
| $61$ |
\( T^{7} + 24 T^{6} + \cdots - 59067 \)
|
| $67$ |
\( T^{7} + 12 T^{6} + \cdots + 6033 \)
|
| $71$ |
\( T^{7} - 20 T^{6} + \cdots - 488617 \)
|
| $73$ |
\( T^{7} + 33 T^{6} + \cdots + 2763 \)
|
| $79$ |
\( T^{7} + 10 T^{6} + \cdots + 387809 \)
|
| $83$ |
\( T^{7} + 25 T^{6} + \cdots + 911 \)
|
| $89$ |
\( T^{7} + 11 T^{6} + \cdots - 24873 \)
|
| $97$ |
\( T^{7} + 51 T^{6} + \cdots + 805725 \)
|
| show more | |
| show less |