Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(2.08498965757\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.168897.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 100x + 256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 100x + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 3\nu - 68 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{2} - 3\beta _1 + 68 \)
|
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 |
|
−8.64858 | 10.2870 | 42.7979 | 92.1784 | −88.9676 | 2.86088 | −93.3863 | −137.178 | −797.212 | |||||||||||||||||||||||||||
| 1.2 | 4.68079 | 13.5120 | −10.0902 | 15.4272 | 63.2466 | 12.5359 | −197.015 | −60.4272 | 72.2115 | ||||||||||||||||||||||||||||
| 1.3 | 10.9678 | −15.7989 | 88.2923 | −51.6056 | −173.279 | −75.3967 | 617.402 | 6.60562 | −565.999 | ||||||||||||||||||||||||||||
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 | 13.6.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 117.6.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 208.6.a.j | 3 | ||
| 5.b | even | 2 | 1 | 325.6.a.c | 3 | ||
| 5.c | odd | 4 | 2 | 325.6.b.c | 6 | ||
| 7.b | odd | 2 | 1 | 637.6.a.b | 3 | ||
| 8.b | even | 2 | 1 | 832.6.a.s | 3 | ||
| 8.d | odd | 2 | 1 | 832.6.a.t | 3 | ||
| 13.b | even | 2 | 1 | 169.6.a.b | 3 | ||
| 13.d | odd | 4 | 2 | 169.6.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.6.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 117.6.a.d | 3 | 3.b | odd | 2 | 1 | ||
| 169.6.a.b | 3 | 13.b | even | 2 | 1 | ||
| 169.6.b.b | 6 | 13.d | odd | 4 | 2 | ||
| 208.6.a.j | 3 | 4.b | odd | 2 | 1 | ||
| 325.6.a.c | 3 | 5.b | even | 2 | 1 | ||
| 325.6.b.c | 6 | 5.c | odd | 4 | 2 | ||
| 637.6.a.b | 3 | 7.b | odd | 2 | 1 | ||
| 832.6.a.s | 3 | 8.b | even | 2 | 1 | ||
| 832.6.a.t | 3 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 7T_{2}^{2} - 84T_{2} + 444 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 7 T^{2} + \cdots + 444 \)
|
| $3$ |
\( T^{3} - 8 T^{2} + \cdots + 2196 \)
|
| $5$ |
\( T^{3} - 56 T^{2} + \cdots + 73386 \)
|
| $7$ |
\( T^{3} + 60 T^{2} + \cdots + 2704 \)
|
| $11$ |
\( T^{3} - 556 T^{2} + \cdots + 39698256 \)
|
| $13$ |
\( (T - 169)^{3} \)
|
| $17$ |
\( T^{3} - 908 T^{2} + \cdots + 77884638 \)
|
| $19$ |
\( T^{3} + \cdots + 1415854512 \)
|
| $23$ |
\( T^{3} + \cdots + 5045833728 \)
|
| $29$ |
\( T^{3} + \cdots - 221025174456 \)
|
| $31$ |
\( T^{3} + \cdots - 1607044480 \)
|
| $37$ |
\( T^{3} + \cdots + 46212896426 \)
|
| $41$ |
\( T^{3} + \cdots + 29456898048 \)
|
| $43$ |
\( T^{3} + \cdots + 281385762060 \)
|
| $47$ |
\( T^{3} + \cdots - 696870885384 \)
|
| $53$ |
\( T^{3} + \cdots - 4415410372608 \)
|
| $59$ |
\( T^{3} + \cdots - 1932677407728 \)
|
| $61$ |
\( T^{3} + \cdots - 18650455523968 \)
|
| $67$ |
\( T^{3} + \cdots + 2080268535536 \)
|
| $71$ |
\( T^{3} + \cdots + 37395101110464 \)
|
| $73$ |
\( T^{3} + \cdots + 5649650834008 \)
|
| $79$ |
\( T^{3} + \cdots - 2044988893184 \)
|
| $83$ |
\( T^{3} + \cdots + 17971240920768 \)
|
| $89$ |
\( T^{3} + \cdots - 28887869991912 \)
|
| $97$ |
\( T^{3} + \cdots - 12102379894216 \)
|
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