Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,4,Mod(3,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.c (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(0.767024830075\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−2.00000 | − | 3.46410i | −1.00000 | − | 1.73205i | −4.00000 | + | 6.92820i | 17.0000 | −4.00000 | + | 6.92820i | −10.0000 | + | 17.3205i | 0 | 11.5000 | − | 19.9186i | −34.0000 | − | 58.8897i | ||||||||||
| 9.1 | −2.00000 | + | 3.46410i | −1.00000 | + | 1.73205i | −4.00000 | − | 6.92820i | 17.0000 | −4.00000 | − | 6.92820i | −10.0000 | − | 17.3205i | 0 | 11.5000 | + | 19.9186i | −34.0000 | + | 58.8897i | |||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 13.4.c.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 117.4.g.c | 2 | ||
| 4.b | odd | 2 | 1 | 208.4.i.b | 2 | ||
| 13.b | even | 2 | 1 | 169.4.c.d | 2 | ||
| 13.c | even | 3 | 1 | inner | 13.4.c.a | ✓ | 2 |
| 13.c | even | 3 | 1 | 169.4.a.d | 1 | ||
| 13.d | odd | 4 | 2 | 169.4.e.c | 4 | ||
| 13.e | even | 6 | 1 | 169.4.a.a | 1 | ||
| 13.e | even | 6 | 1 | 169.4.c.d | 2 | ||
| 13.f | odd | 12 | 2 | 169.4.b.c | 2 | ||
| 13.f | odd | 12 | 2 | 169.4.e.c | 4 | ||
| 39.h | odd | 6 | 1 | 1521.4.a.k | 1 | ||
| 39.i | odd | 6 | 1 | 117.4.g.c | 2 | ||
| 39.i | odd | 6 | 1 | 1521.4.a.b | 1 | ||
| 52.j | odd | 6 | 1 | 208.4.i.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.4.c.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 13.4.c.a | ✓ | 2 | 13.c | even | 3 | 1 | inner |
| 117.4.g.c | 2 | 3.b | odd | 2 | 1 | ||
| 117.4.g.c | 2 | 39.i | odd | 6 | 1 | ||
| 169.4.a.a | 1 | 13.e | even | 6 | 1 | ||
| 169.4.a.d | 1 | 13.c | even | 3 | 1 | ||
| 169.4.b.c | 2 | 13.f | odd | 12 | 2 | ||
| 169.4.c.d | 2 | 13.b | even | 2 | 1 | ||
| 169.4.c.d | 2 | 13.e | even | 6 | 1 | ||
| 169.4.e.c | 4 | 13.d | odd | 4 | 2 | ||
| 169.4.e.c | 4 | 13.f | odd | 12 | 2 | ||
| 208.4.i.b | 2 | 4.b | odd | 2 | 1 | ||
| 208.4.i.b | 2 | 52.j | odd | 6 | 1 | ||
| 1521.4.a.b | 1 | 39.i | odd | 6 | 1 | ||
| 1521.4.a.k | 1 | 39.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4T_{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4T + 16 \)
|
| $3$ |
\( T^{2} + 2T + 4 \)
|
| $5$ |
\( (T - 17)^{2} \)
|
| $7$ |
\( T^{2} + 20T + 400 \)
|
| $11$ |
\( T^{2} - 32T + 1024 \)
|
| $13$ |
\( T^{2} + 91T + 2197 \)
|
| $17$ |
\( T^{2} - 13T + 169 \)
|
| $19$ |
\( T^{2} + 30T + 900 \)
|
| $23$ |
\( T^{2} + 78T + 6084 \)
|
| $29$ |
\( T^{2} + 197T + 38809 \)
|
| $31$ |
\( (T + 74)^{2} \)
|
| $37$ |
\( T^{2} - 227T + 51529 \)
|
| $41$ |
\( T^{2} - 165T + 27225 \)
|
| $43$ |
\( T^{2} - 156T + 24336 \)
|
| $47$ |
\( (T + 162)^{2} \)
|
| $53$ |
\( (T - 93)^{2} \)
|
| $59$ |
\( T^{2} - 864T + 746496 \)
|
| $61$ |
\( T^{2} + 145T + 21025 \)
|
| $67$ |
\( T^{2} + 862T + 743044 \)
|
| $71$ |
\( T^{2} + 654T + 427716 \)
|
| $73$ |
\( (T - 215)^{2} \)
|
| $79$ |
\( (T + 76)^{2} \)
|
| $83$ |
\( (T - 628)^{2} \)
|
| $89$ |
\( T^{2} - 266T + 70756 \)
|
| $97$ |
\( T^{2} + 238T + 56644 \)
|
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