Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [731,2,Mod(1,731)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, 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("731.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 731 = 17 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 731.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: | \(5.83706438776\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 9x^{6} + 9x^{5} + 21x^{4} - 21x^{3} - 8x^{2} + 7x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 9x^{6} + 9x^{5} + 21x^{4} - 21x^{3} - 8x^{2} + 7x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 9\nu^{5} + 9\nu^{4} + 21\nu^{3} - 20\nu^{2} - 8\nu + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 9\nu^{5} + 9\nu^{4} + 21\nu^{3} - 21\nu^{2} - 8\nu + 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{7} - \nu^{6} - 18\nu^{5} + 9\nu^{4} + 43\nu^{3} - 21\nu^{2} - 22\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -4\nu^{7} + 3\nu^{6} + 37\nu^{5} - 27\nu^{4} - 92\nu^{3} + 62\nu^{2} + 48\nu - 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( 4\nu^{7} - 3\nu^{6} - 37\nu^{5} + 27\nu^{4} + 93\nu^{3} - 62\nu^{2} - 52\nu + 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( -5\nu^{7} + 4\nu^{6} + 46\nu^{5} - 35\nu^{4} - 114\nu^{3} + 77\nu^{2} + 60\nu - 18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 5\beta_{3} + 6\beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{6} + 8\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 18\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{7} + 8\beta_{6} - \beta_{5} + \beta_{4} - 26\beta_{3} + 33\beta_{2} + 2\beta _1 + 37 \)
|
| \(\nu^{7}\) | \(=\) |
\( 41\beta_{6} + 50\beta_{5} + 10\beta_{4} + 8\beta_{3} + 9\beta_{2} + 88\beta _1 + 10 \)
|
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 |
|
−2.31200 | −1.95532 | 3.34533 | 1.83479 | 4.52069 | 0.437329 | −3.11039 | 0.823276 | −4.24203 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.78724 | 1.17799 | 1.19423 | −1.95027 | −2.10535 | −1.57661 | 1.44011 | −1.61234 | 3.48560 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.700276 | −2.17405 | −1.50961 | −0.553685 | 1.52243 | −0.939298 | 2.45770 | 1.72649 | 0.387732 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.235409 | 2.48067 | −1.94458 | −0.468153 | 0.583972 | −5.07261 | −0.928590 | 3.15374 | −0.110207 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.298156 | 0.0958345 | −1.91110 | 0.959960 | 0.0285736 | 1.32998 | −1.16612 | −2.99082 | 0.286218 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.26872 | 1.44891 | −0.390345 | −3.92944 | 1.83826 | 0.416048 | −3.03268 | −0.900661 | −4.98536 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.66158 | −3.15537 | 0.760849 | 0.0956277 | −5.24290 | 3.17624 | −2.05895 | 6.95637 | 0.158893 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.33565 | −0.918667 | 3.45524 | −2.98883 | −2.14568 | −2.77109 | 3.39893 | −2.15605 | −6.98085 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 731.2.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 6579.2.a.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 731.2.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 6579.2.a.k | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} - 9T_{2}^{6} + 9T_{2}^{5} + 21T_{2}^{4} - 21T_{2}^{3} - 8T_{2}^{2} + 7T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 9 T^{6} + \cdots - 1 \)
|
| $3$ |
\( T^{8} + 3 T^{7} + \cdots + 5 \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots - 1 \)
|
| $7$ |
\( T^{8} + 5 T^{7} + \cdots + 16 \)
|
| $11$ |
\( T^{8} + 4 T^{7} + \cdots - 4040 \)
|
| $13$ |
\( T^{8} + 12 T^{7} + \cdots - 73 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} - 36 T^{6} + \cdots - 584 \)
|
| $23$ |
\( T^{8} + 9 T^{7} + \cdots - 304 \)
|
| $29$ |
\( T^{8} + 27 T^{7} + \cdots + 508 \)
|
| $31$ |
\( T^{8} + 12 T^{7} + \cdots + 316 \)
|
| $37$ |
\( T^{8} + 24 T^{7} + \cdots + 59777 \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots - 184 \)
|
| $43$ |
\( (T - 1)^{8} \)
|
| $47$ |
\( T^{8} - 15 T^{7} + \cdots - 2575 \)
|
| $53$ |
\( T^{8} + 23 T^{7} + \cdots + 1011811 \)
|
| $59$ |
\( T^{8} - 16 T^{7} + \cdots + 1 \)
|
| $61$ |
\( T^{8} + 34 T^{7} + \cdots - 411065 \)
|
| $67$ |
\( T^{8} - 219 T^{6} + \cdots + 34597 \)
|
| $71$ |
\( T^{8} + 3 T^{7} + \cdots - 378151 \)
|
| $73$ |
\( T^{8} + 3 T^{7} + \cdots - 59 \)
|
| $79$ |
\( T^{8} + 24 T^{7} + \cdots - 25708 \)
|
| $83$ |
\( T^{8} + 8 T^{7} + \cdots + 1237952 \)
|
| $89$ |
\( T^{8} - 23 T^{7} + \cdots + 5884640 \)
|
| $97$ |
\( T^{8} + 10 T^{7} + \cdots + 12078056 \)
|
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