Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,18,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, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| 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: | \(142.913228129\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 13143327x^{2} + 23482672141x - 10722018405934 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{3}\cdot 53 \) |
| 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 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
\( x^{4} - x^{3} - 13143327x^{2} + 23482672141x - 10722018405934 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 1276\nu^{2} - 10781779\nu + 9219399110 ) / 2016 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 1028\nu^{2} + 19739731\nu - 24362751878 ) / 5040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -11\nu^{3} - 17492\nu^{2} + 117984401\nu - 78701566498 ) / 10080 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{3} + 3\beta_{2} + 10\beta _1 + 1272 ) / 5088 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2592\beta_{3} - 89\beta_{2} - 5738\beta _1 + 5572770860 ) / 848 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 62971468\beta_{3} + 33026721\beta_{2} + 162005326\beta _1 - 89559721952952 ) / 5088 \)
|
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 |
|
256.000 | −6561.00 | 65536.0 | −1.12863e6 | −1.67962e6 | −2.01146e7 | 1.67772e7 | 4.30467e7 | −2.88929e8 | ||||||||||||||||||||||||||||||
| 1.2 | 256.000 | −6561.00 | 65536.0 | −805857. | −1.67962e6 | −4.19311e6 | 1.67772e7 | 4.30467e7 | −2.06299e8 | |||||||||||||||||||||||||||||||
| 1.3 | 256.000 | −6561.00 | 65536.0 | 220653. | −1.67962e6 | 1.26874e7 | 1.67772e7 | 4.30467e7 | 5.64871e7 | |||||||||||||||||||||||||||||||
| 1.4 | 256.000 | −6561.00 | 65536.0 | 1.19525e6 | −1.67962e6 | −5.20600e6 | 1.67772e7 | 4.30467e7 | 3.05983e8 | |||||||||||||||||||||||||||||||
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 | 78.18.a.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.18.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 518584T_{5}^{3} - 1565790671060T_{5}^{2} - 777588880936020000T_{5} + 239869915320770184600000 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(78))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 256)^{4} \)
|
| $3$ |
\( (T + 6561)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 55\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots - 51\!\cdots\!20 \)
|
| $13$ |
\( (T - 815730721)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 10\!\cdots\!60 \)
|
| $19$ |
\( T^{4} + \cdots + 19\!\cdots\!92 \)
|
| $23$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots - 16\!\cdots\!88 \)
|
| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!44 \)
|
| $37$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 36\!\cdots\!96 \)
|
| $43$ |
\( T^{4} + \cdots - 66\!\cdots\!72 \)
|
| $47$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 68\!\cdots\!72 \)
|
| $71$ |
\( T^{4} + \cdots + 13\!\cdots\!64 \)
|
| $73$ |
\( T^{4} + \cdots - 29\!\cdots\!60 \)
|
| $79$ |
\( T^{4} + \cdots + 32\!\cdots\!76 \)
|
| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 63\!\cdots\!08 \)
|
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