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: |
\( x^{4} - 2x^{3} - 46381379x^{2} - 41728285884x + 168961087598724 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{5}\cdot 13 \) |
| 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
\( x^{4} - 2x^{3} - 46381379x^{2} - 41728285884x + 168961087598724 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -37\nu^{3} + 32707\nu^{2} + 1422123276\nu + 401325124068 ) / 3157792 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 15775\nu^{2} + 8596280\nu - 459934800228 ) / 1578896 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 6703\nu^{2} - 19227156\nu + 124091025516 ) / 394724 \)
|
| \(\nu\) | \(=\) |
\( ( 19\beta_{3} + 24\beta_{2} + 8\beta _1 + 1404 ) / 2808 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -127757\beta_{3} + 79224\beta_{2} - 14776\beta _1 + 65119458924 ) / 2808 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 617345785\beta_{3} + 992490216\beta_{2} + 54773720\beta _1 + 88075128445668 ) / 2808 \)
|
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 | −1.29173e6 | 1.67962e6 | −5.08513e6 | 1.67772e7 | 4.30467e7 | −3.30682e8 | ||||||||||||||||||||||||||||||
| 1.2 | 256.000 | 6561.00 | 65536.0 | −403785. | 1.67962e6 | 1.62547e7 | 1.67772e7 | 4.30467e7 | −1.03369e8 | |||||||||||||||||||||||||||||||
| 1.3 | 256.000 | 6561.00 | 65536.0 | 309126. | 1.67962e6 | −1.09867e7 | 1.67772e7 | 4.30467e7 | 7.91364e7 | |||||||||||||||||||||||||||||||
| 1.4 | 256.000 | 6561.00 | 65536.0 | 419036. | 1.67962e6 | −1.06738e7 | 1.67772e7 | 4.30467e7 | 1.07273e8 | |||||||||||||||||||||||||||||||
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.f | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.18.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 967348T_{5}^{3} - 583492416024T_{5}^{2} - 160166478250522400T_{5} + 67562854182643803280000 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(78))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 256)^{4} \)
|
| $3$ |
\( (T - 6561)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 67\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 96\!\cdots\!56 \)
|
| $11$ |
\( T^{4} + \cdots + 23\!\cdots\!16 \)
|
| $13$ |
\( (T + 815730721)^{4} \)
|
| $17$ |
\( T^{4} + \cdots - 26\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 60\!\cdots\!64 \)
|
| $23$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 12\!\cdots\!12 \)
|
| $31$ |
\( T^{4} + \cdots - 55\!\cdots\!32 \)
|
| $37$ |
\( T^{4} + \cdots + 47\!\cdots\!08 \)
|
| $41$ |
\( T^{4} + \cdots + 15\!\cdots\!40 \)
|
| $43$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots - 21\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 63\!\cdots\!40 \)
|
| $59$ |
\( T^{4} + \cdots - 69\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots - 88\!\cdots\!52 \)
|
| $67$ |
\( T^{4} + \cdots + 67\!\cdots\!72 \)
|
| $71$ |
\( T^{4} + \cdots + 49\!\cdots\!36 \)
|
| $73$ |
\( T^{4} + \cdots - 99\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 45\!\cdots\!92 \)
|
| $83$ |
\( T^{4} + \cdots + 16\!\cdots\!92 \)
|
| $89$ |
\( T^{4} + \cdots - 30\!\cdots\!48 \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
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