Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,13,Mod(5,6)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.5");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6.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: | \(5.48396290366\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{1009})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 499x^{2} + 500x + 64518 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{5} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 499x^{2} + 500x + 64518 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 64\nu^{3} - 96\nu^{2} - 15712\nu + 7872 ) / 1017 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -88\nu^{3} - 3936\nu^{2} + 66352\nu + 985836 ) / 339 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -10\nu^{3} + 15270\nu^{2} - 596\nu - 3821082 ) / 1017 \)
|
| \(\nu\) | \(=\) |
\( ( 16\beta_{3} + 20\beta_{2} + 85\beta _1 + 1296 ) / 2592 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 44\beta_{3} + \beta_{2} + 11\beta _1 + 162324 ) / 648 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2096\beta_{3} + 2458\beta_{2} + 31061\beta _1 + 486648 ) / 1296 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
− | 45.2548i | 4.41144 | − | 728.987i | −2048.00 | 1793.38i | −32990.2 | − | 199.639i | −136690. | 92681.9i | −531402. | − | 6431.76i | 81159.0 | |||||||||||||||||||||||
| 5.2 | − | 45.2548i | 385.589 | + | 618.678i | −2048.00 | 16348.2i | 27998.2 | − | 17449.7i | 213230. | 92681.9i | −234084. | + | 477110.i | 739833. | ||||||||||||||||||||||||
| 5.3 | 45.2548i | 4.41144 | + | 728.987i | −2048.00 | − | 1793.38i | −32990.2 | + | 199.639i | −136690. | − | 92681.9i | −531402. | + | 6431.76i | 81159.0 | |||||||||||||||||||||||
| 5.4 | 45.2548i | 385.589 | − | 618.678i | −2048.00 | − | 16348.2i | 27998.2 | + | 17449.7i | 213230. | − | 92681.9i | −234084. | − | 477110.i | 739833. | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6.13.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 6.13.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 48.13.e.c | 4 | ||
| 5.b | even | 2 | 1 | 150.13.d.a | 4 | ||
| 5.c | odd | 4 | 2 | 150.13.b.a | 8 | ||
| 8.b | even | 2 | 1 | 192.13.e.e | 4 | ||
| 8.d | odd | 2 | 1 | 192.13.e.h | 4 | ||
| 9.c | even | 3 | 2 | 162.13.d.d | 8 | ||
| 9.d | odd | 6 | 2 | 162.13.d.d | 8 | ||
| 12.b | even | 2 | 1 | 48.13.e.c | 4 | ||
| 15.d | odd | 2 | 1 | 150.13.d.a | 4 | ||
| 15.e | even | 4 | 2 | 150.13.b.a | 8 | ||
| 24.f | even | 2 | 1 | 192.13.e.h | 4 | ||
| 24.h | odd | 2 | 1 | 192.13.e.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.13.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 6.13.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 48.13.e.c | 4 | 4.b | odd | 2 | 1 | ||
| 48.13.e.c | 4 | 12.b | even | 2 | 1 | ||
| 150.13.b.a | 8 | 5.c | odd | 4 | 2 | ||
| 150.13.b.a | 8 | 15.e | even | 4 | 2 | ||
| 150.13.d.a | 4 | 5.b | even | 2 | 1 | ||
| 150.13.d.a | 4 | 15.d | odd | 2 | 1 | ||
| 162.13.d.d | 8 | 9.c | even | 3 | 2 | ||
| 162.13.d.d | 8 | 9.d | odd | 6 | 2 | ||
| 192.13.e.e | 4 | 8.b | even | 2 | 1 | ||
| 192.13.e.e | 4 | 24.h | odd | 2 | 1 | ||
| 192.13.e.h | 4 | 8.d | odd | 2 | 1 | ||
| 192.13.e.h | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(6, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2048)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 282429536481 \)
|
| $5$ |
\( T^{4} + \cdots + 859568578560000 \)
|
| $7$ |
\( (T^{2} - 76540 T - 29146513676)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 22\!\cdots\!84 \)
|
| $13$ |
\( (T^{2} + \cdots - 23192320437500)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 36\!\cdots\!04 \)
|
| $19$ |
\( (T^{2} + \cdots - 8399894140076)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 61\!\cdots\!04 \)
|
| $29$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots + 26\!\cdots\!24)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 11\!\cdots\!96)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 40\!\cdots\!84 \)
|
| $43$ |
\( (T^{2} + \cdots - 32\!\cdots\!44)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 40\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 68\!\cdots\!04 \)
|
| $61$ |
\( (T^{2} + \cdots - 52\!\cdots\!24)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 28\!\cdots\!64)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 72\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 51\!\cdots\!36)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 22\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 16\!\cdots\!24 \)
|
| $97$ |
\( (T^{2} + \cdots - 10\!\cdots\!24)^{2} \)
|
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