Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,6,Mod(12,13)]
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([1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.12");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.b (of order \(2\), degree \(1\), 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: | \(2.08498965757\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 161x^{4} + 5856x^{2} + 18864 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 161x^{4} + 5856x^{2} + 18864 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 101\nu^{2} + 660 ) / 72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 173\nu^{3} + 6492\nu ) / 144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 125\nu^{3} + 2316\nu ) / 48 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} + 173\nu^{2} + 4548 ) / 72 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{2} - 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + 3\beta_{3} - 87\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -101\beta_{5} + 173\beta_{2} + 4794 \)
|
| \(\nu^{5}\) | \(=\) |
\( 173\beta_{4} - 375\beta_{3} + 8559\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 12.1 |
|
− | 10.4280i | 23.8626 | −76.7440 | 46.3074i | − | 248.840i | 16.0658i | 466.592i | 326.425 | 482.896 | ||||||||||||||||||||||||||||||||||
| 12.2 | − | 6.97807i | −23.2081 | −16.6935 | − | 14.2993i | 161.948i | − | 181.853i | − | 106.810i | 295.618 | −99.7813 | |||||||||||||||||||||||||||||||||
| 12.3 | − | 1.88746i | 7.34551 | 28.4375 | − | 75.2938i | − | 13.8644i | 222.759i | − | 114.074i | −189.043 | −142.114 | |||||||||||||||||||||||||||||||||
| 12.4 | 1.88746i | 7.34551 | 28.4375 | 75.2938i | 13.8644i | − | 222.759i | 114.074i | −189.043 | −142.114 | ||||||||||||||||||||||||||||||||||||
| 12.5 | 6.97807i | −23.2081 | −16.6935 | 14.2993i | − | 161.948i | 181.853i | 106.810i | 295.618 | −99.7813 | ||||||||||||||||||||||||||||||||||||
| 12.6 | 10.4280i | 23.8626 | −76.7440 | − | 46.3074i | 248.840i | − | 16.0658i | − | 466.592i | 326.425 | 482.896 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 13.6.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 117.6.b.c | 6 | ||
| 4.b | odd | 2 | 1 | 208.6.f.c | 6 | ||
| 13.b | even | 2 | 1 | inner | 13.6.b.a | ✓ | 6 |
| 13.d | odd | 4 | 2 | 169.6.a.e | 6 | ||
| 39.d | odd | 2 | 1 | 117.6.b.c | 6 | ||
| 52.b | odd | 2 | 1 | 208.6.f.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.6.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 13.6.b.a | ✓ | 6 | 13.b | even | 2 | 1 | inner |
| 117.6.b.c | 6 | 3.b | odd | 2 | 1 | ||
| 117.6.b.c | 6 | 39.d | odd | 2 | 1 | ||
| 169.6.a.e | 6 | 13.d | odd | 4 | 2 | ||
| 208.6.f.c | 6 | 4.b | odd | 2 | 1 | ||
| 208.6.f.c | 6 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(13, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 161 T^{4} + \cdots + 18864 \)
|
| $3$ |
\( (T^{3} - 8 T^{2} + \cdots + 4068)^{2} \)
|
| $5$ |
\( T^{6} + \cdots + 2485690416 \)
|
| $7$ |
\( T^{6} + \cdots + 423560602764 \)
|
| $11$ |
\( T^{6} + \cdots + 7521473396736 \)
|
| $13$ |
\( T^{6} + \cdots + 51\!\cdots\!57 \)
|
| $17$ |
\( (T^{3} + 696 T^{2} + \cdots + 227049966)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 43\!\cdots\!16 \)
|
| $23$ |
\( (T^{3} + 4752 T^{2} + \cdots + 1744071264)^{2} \)
|
| $29$ |
\( (T^{3} - 7518 T^{2} + \cdots + 1625868288)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 52\!\cdots\!56 \)
|
| $37$ |
\( T^{6} + \cdots + 20\!\cdots\!84 \)
|
| $41$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $43$ |
\( (T^{3} - 12792 T^{2} + \cdots + 191268739948)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 26\!\cdots\!84 \)
|
| $53$ |
\( (T^{3} + \cdots - 1324868101272)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 14\!\cdots\!44 \)
|
| $61$ |
\( (T^{3} + \cdots - 21319109288576)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 98\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots + 22\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots + 30\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} + \cdots + 3525550887232)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 79\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 44\!\cdots\!44 \)
|
| $97$ |
\( T^{6} + \cdots + 46\!\cdots\!36 \)
|
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