Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,8,Mod(1,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.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: | \(24.3660319878\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{10}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(\beta = 6\sqrt{10}\). 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 |
|
8.00000 | −27.0000 | 64.0000 | −168.710 | −216.000 | 501.710 | 512.000 | 729.000 | −1349.68 | ||||||||||||||||||||||||
| 1.2 | 8.00000 | −27.0000 | 64.0000 | 248.710 | −216.000 | −1357.71 | 512.000 | 729.000 | 1989.68 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(13\) | \(1\) |
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 | 78.8.a.g | ✓ | 2 |
| 3.b | odd | 2 | 1 | 234.8.a.h | 2 | ||
| 4.b | odd | 2 | 1 | 624.8.a.j | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.8.a.g | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 234.8.a.h | 2 | 3.b | odd | 2 | 1 | ||
| 624.8.a.j | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 80T_{5} - 41960 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(78))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{2} \)
|
| $3$ |
\( (T + 27)^{2} \)
|
| $5$ |
\( T^{2} - 80T - 41960 \)
|
| $7$ |
\( T^{2} + 856T - 681176 \)
|
| $11$ |
\( T^{2} + 2540 T - 358460 \)
|
| $13$ |
\( (T + 2197)^{2} \)
|
| $17$ |
\( T^{2} + 42540 T + 360239940 \)
|
| $19$ |
\( T^{2} + \cdots - 2005070040 \)
|
| $23$ |
\( T^{2} + \cdots - 9570139760 \)
|
| $29$ |
\( T^{2} + \cdots + 22183039204 \)
|
| $31$ |
\( T^{2} + \cdots + 79387174936 \)
|
| $37$ |
\( T^{2} + \cdots - 69891567804 \)
|
| $41$ |
\( T^{2} + \cdots - 363703035816 \)
|
| $43$ |
\( T^{2} + \cdots + 7134321136 \)
|
| $47$ |
\( T^{2} + \cdots - 1229165045276 \)
|
| $53$ |
\( T^{2} + \cdots - 103227295356 \)
|
| $59$ |
\( T^{2} + \cdots - 923004059324 \)
|
| $61$ |
\( T^{2} + \cdots + 4129001509924 \)
|
| $67$ |
\( T^{2} + \cdots - 6944774801000 \)
|
| $71$ |
\( T^{2} + \cdots - 2653359383996 \)
|
| $73$ |
\( T^{2} + \cdots + 3981643076164 \)
|
| $79$ |
\( T^{2} + \cdots - 34229994711104 \)
|
| $83$ |
\( T^{2} + \cdots + 3723228979524 \)
|
| $89$ |
\( T^{2} + \cdots + 101563294611816 \)
|
| $97$ |
\( T^{2} + \cdots - 87915642889244 \)
|
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