Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,10,Mod(1,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.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: | \(6.69546587013\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 55\nu^{3} + 317\nu^{2} - 39383\nu + 189604 ) / 1088 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{4} - 113\nu^{3} - 6571\nu^{2} + 54545\nu + 495172 ) / 1088 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} - 4\nu - 575 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 4\beta _1 + 575 \)
|
| \(\nu^{3}\) | \(=\) |
\( 16\beta_{4} + 4\beta_{3} + 28\beta_{2} + 877\beta _1 + 2500 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1197\beta_{4} + 220\beta_{3} + 452\beta_{2} + 10120\beta _1 + 509379 \)
|
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 |
|
−24.7188 | 194.269 | 99.0182 | −920.299 | −4802.09 | 5359.02 | 10208.4 | 18057.4 | 22748.7 | |||||||||||||||||||||||||||||||||
| 1.2 | −21.3176 | −195.094 | −57.5590 | −1277.14 | 4158.93 | −2277.98 | 12141.6 | 18378.5 | 27225.6 | ||||||||||||||||||||||||||||||||||
| 1.3 | 3.15034 | −136.532 | −502.075 | 2554.62 | −430.124 | 9399.91 | −3194.68 | −1041.89 | 8047.92 | ||||||||||||||||||||||||||||||||||
| 1.4 | 19.7176 | 250.479 | −123.217 | 1555.58 | 4938.84 | −8329.39 | −12524.9 | 43056.6 | 30672.3 | ||||||||||||||||||||||||||||||||||
| 1.5 | 38.1685 | 47.8784 | 944.833 | −109.762 | 1827.45 | 5947.44 | 16520.6 | −17390.7 | −4189.45 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
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 | 13.10.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 117.10.a.e | 5 | ||
| 4.b | odd | 2 | 1 | 208.10.a.h | 5 | ||
| 5.b | even | 2 | 1 | 325.10.a.b | 5 | ||
| 13.b | even | 2 | 1 | 169.10.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.10.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 117.10.a.e | 5 | 3.b | odd | 2 | 1 | ||
| 169.10.a.b | 5 | 13.b | even | 2 | 1 | ||
| 208.10.a.h | 5 | 4.b | odd | 2 | 1 | ||
| 325.10.a.b | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 15T_{2}^{4} - 1348T_{2}^{3} + 8508T_{2}^{2} + 383520T_{2} - 1249344 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 15 T^{4} + \cdots - 1249344 \)
|
| $3$ |
\( T^{5} + \cdots - 62057286864 \)
|
| $5$ |
\( T^{5} + \cdots + 512670311383500 \)
|
| $7$ |
\( T^{5} + \cdots - 56\!\cdots\!88 \)
|
| $11$ |
\( T^{5} + \cdots - 19\!\cdots\!96 \)
|
| $13$ |
\( (T - 28561)^{5} \)
|
| $17$ |
\( T^{5} + \cdots - 39\!\cdots\!92 \)
|
| $19$ |
\( T^{5} + \cdots - 29\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 42\!\cdots\!52 \)
|
| $29$ |
\( T^{5} + \cdots - 44\!\cdots\!20 \)
|
| $31$ |
\( T^{5} + \cdots + 48\!\cdots\!64 \)
|
| $37$ |
\( T^{5} + \cdots - 51\!\cdots\!48 \)
|
| $41$ |
\( T^{5} + \cdots + 49\!\cdots\!12 \)
|
| $43$ |
\( T^{5} + \cdots + 16\!\cdots\!08 \)
|
| $47$ |
\( T^{5} + \cdots - 10\!\cdots\!64 \)
|
| $53$ |
\( T^{5} + \cdots - 36\!\cdots\!12 \)
|
| $59$ |
\( T^{5} + \cdots - 43\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 57\!\cdots\!08 \)
|
| $67$ |
\( T^{5} + \cdots - 85\!\cdots\!64 \)
|
| $71$ |
\( T^{5} + \cdots - 41\!\cdots\!92 \)
|
| $73$ |
\( T^{5} + \cdots + 16\!\cdots\!16 \)
|
| $79$ |
\( T^{5} + \cdots - 41\!\cdots\!40 \)
|
| $83$ |
\( T^{5} + \cdots - 52\!\cdots\!24 \)
|
| $89$ |
\( T^{5} + \cdots - 19\!\cdots\!60 \)
|
| $97$ |
\( T^{5} + \cdots - 14\!\cdots\!32 \)
|
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