Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [56,6,Mod(1,56)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("56.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 56 = 2^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 56.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: | \(8.98149390953\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{345}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 86 \)
|
| 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 \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
0 | −21.5742 | 0 | −14.7225 | 0 | −49.0000 | 0 | 222.445 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 15.5742 | 0 | 96.7225 | 0 | −49.0000 | 0 | −0.445054 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
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 | 56.6.a.e | ✓ | 2 |
| 3.b | odd | 2 | 1 | 504.6.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 112.6.a.i | 2 | ||
| 7.b | odd | 2 | 1 | 392.6.a.d | 2 | ||
| 7.c | even | 3 | 2 | 392.6.i.j | 4 | ||
| 7.d | odd | 6 | 2 | 392.6.i.i | 4 | ||
| 8.b | even | 2 | 1 | 448.6.a.v | 2 | ||
| 8.d | odd | 2 | 1 | 448.6.a.t | 2 | ||
| 12.b | even | 2 | 1 | 1008.6.a.bd | 2 | ||
| 28.d | even | 2 | 1 | 784.6.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.6.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 112.6.a.i | 2 | 4.b | odd | 2 | 1 | ||
| 392.6.a.d | 2 | 7.b | odd | 2 | 1 | ||
| 392.6.i.i | 4 | 7.d | odd | 6 | 2 | ||
| 392.6.i.j | 4 | 7.c | even | 3 | 2 | ||
| 448.6.a.t | 2 | 8.d | odd | 2 | 1 | ||
| 448.6.a.v | 2 | 8.b | even | 2 | 1 | ||
| 504.6.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 784.6.a.u | 2 | 28.d | even | 2 | 1 | ||
| 1008.6.a.bd | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 6T_{3} - 336 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(56))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 6T - 336 \)
|
| $5$ |
\( T^{2} - 82T - 1424 \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( T^{2} - 340T + 16480 \)
|
| $13$ |
\( T^{2} - 910T - 317720 \)
|
| $17$ |
\( T^{2} - 3216 T + 2573244 \)
|
| $19$ |
\( T^{2} + 674T - 585056 \)
|
| $23$ |
\( T^{2} + 1104 T - 1483776 \)
|
| $29$ |
\( T^{2} - 8064 T + 11773404 \)
|
| $31$ |
\( T^{2} + 6212 T - 5567264 \)
|
| $37$ |
\( T^{2} + 8512 T - 34360964 \)
|
| $41$ |
\( T^{2} + 1304 T - 158165876 \)
|
| $43$ |
\( T^{2} + 10004 T - 194677376 \)
|
| $47$ |
\( T^{2} + 12748 T - 135251744 \)
|
| $53$ |
\( T^{2} + 11220 T - 172017180 \)
|
| $59$ |
\( T^{2} + \cdots - 1715115024 \)
|
| $61$ |
\( T^{2} + \cdots + 2636833536 \)
|
| $67$ |
\( T^{2} + \cdots - 1158165296 \)
|
| $71$ |
\( T^{2} - 89720 T + 967897600 \)
|
| $73$ |
\( T^{2} + 55588 T - 243399884 \)
|
| $79$ |
\( T^{2} - 48824 T - 724002176 \)
|
| $83$ |
\( T^{2} + \cdots - 3779136224 \)
|
| $89$ |
\( T^{2} + \cdots - 7628401980 \)
|
| $97$ |
\( T^{2} + \cdots + 1150801884 \)
|
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