Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(4.06100533129\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 354x^{2} - 640x + 20912 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 354x^{2} - 640x + 20912 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 15\nu^{2} - 180\nu + 1956 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 13\nu^{2} - 188\nu + 1606 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2\beta_{2} + 4\beta _1 + 175 \)
|
| \(\nu^{3}\) | \(=\) |
\( 15\beta_{3} - 26\beta_{2} + 240\beta _1 + 669 \)
|
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 |
|
−14.6191 | −51.4405 | 85.7173 | 123.659 | 752.012 | 559.667 | 618.134 | 459.121 | −1807.78 | ||||||||||||||||||||||||||||||
| 1.2 | −3.26058 | 66.7551 | −117.369 | 259.973 | −217.660 | 1453.99 | 800.043 | 2269.24 | −847.662 | |||||||||||||||||||||||||||||||
| 1.3 | 16.1994 | 71.3782 | 134.421 | −532.467 | 1156.29 | −6.33667 | 104.021 | 2907.85 | −8625.66 | |||||||||||||||||||||||||||||||
| 1.4 | 16.6802 | −6.69280 | 150.230 | 406.835 | −111.637 | −315.324 | 370.802 | −2142.21 | 6786.11 | |||||||||||||||||||||||||||||||
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 | 13.8.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 117.8.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 208.8.a.k | 4 | ||
| 5.b | even | 2 | 1 | 325.8.a.c | 4 | ||
| 13.b | even | 2 | 1 | 169.8.a.c | 4 | ||
| 13.d | odd | 4 | 2 | 169.8.b.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.8.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 117.8.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 169.8.a.c | 4 | 13.b | even | 2 | 1 | ||
| 169.8.b.c | 8 | 13.d | odd | 4 | 2 | ||
| 208.8.a.k | 4 | 4.b | odd | 2 | 1 | ||
| 325.8.a.c | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 15T_{2}^{3} - 270T_{2}^{2} + 3264T_{2} + 12880 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 15 T^{3} + \cdots + 12880 \)
|
| $3$ |
\( T^{4} - 80 T^{3} + \cdots + 1640448 \)
|
| $5$ |
\( T^{4} + \cdots - 6964113500 \)
|
| $7$ |
\( T^{4} + \cdots + 1625961532 \)
|
| $11$ |
\( T^{4} + \cdots + 19773464676784 \)
|
| $13$ |
\( (T + 2197)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 17\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 43\!\cdots\!60 \)
|
| $23$ |
\( T^{4} + \cdots + 20\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots - 32\!\cdots\!40 \)
|
| $31$ |
\( T^{4} + \cdots - 83\!\cdots\!08 \)
|
| $37$ |
\( T^{4} + \cdots + 59\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots + 19\!\cdots\!36 \)
|
| $43$ |
\( T^{4} + \cdots + 11\!\cdots\!52 \)
|
| $47$ |
\( T^{4} + \cdots - 13\!\cdots\!60 \)
|
| $53$ |
\( T^{4} + \cdots - 28\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots - 44\!\cdots\!20 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!68 \)
|
| $67$ |
\( T^{4} + \cdots - 57\!\cdots\!08 \)
|
| $71$ |
\( T^{4} + \cdots + 25\!\cdots\!48 \)
|
| $73$ |
\( T^{4} + \cdots + 69\!\cdots\!24 \)
|
| $79$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 70\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots - 76\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 44\!\cdots\!60 \)
|
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