Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [114,12,Mod(1,114)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(114, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("114.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.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: | \(87.5911225838\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - 2x^{3} - 2999302x^{2} - 1003379071x + 558664856310 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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 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
\( x^{4} - 2x^{3} - 2999302x^{2} - 1003379071x + 558664856310 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4626\nu^{3} + 2129602\nu^{2} + 12954384878\nu + 302026698798 ) / 259874153 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -360\nu^{3} + 21273\nu^{2} + 409358265\nu + 240443759281 ) / 37124879 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 426\nu^{3} - 1881417\nu^{2} + 137434443\nu + 2498905881648 ) / 259874153 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} - 5\beta_{2} + 3\beta _1 + 49 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -12435\beta_{3} - 3947\beta_{2} + 1005\beta _1 + 143968063 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1338257\beta_{3} - 7909367\beta_{2} + 1735353\beta _1 + 36340683403 ) / 48 \)
|
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 |
|
−32.0000 | 243.000 | 1024.00 | −6694.47 | −7776.00 | −32315.4 | −32768.0 | 59049.0 | 214223. | ||||||||||||||||||||||||||||||
| 1.2 | −32.0000 | 243.000 | 1024.00 | −2423.65 | −7776.00 | 49322.4 | −32768.0 | 59049.0 | 77557.0 | |||||||||||||||||||||||||||||||
| 1.3 | −32.0000 | 243.000 | 1024.00 | 6182.37 | −7776.00 | −34779.2 | −32768.0 | 59049.0 | −197836. | |||||||||||||||||||||||||||||||
| 1.4 | −32.0000 | 243.000 | 1024.00 | 12847.8 | −7776.00 | 54006.1 | −32768.0 | 59049.0 | −411128. | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 114.12.a.f | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.12.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 9912T_{5}^{3} - 77864355T_{5}^{2} + 415483412850T_{5} + 1288751341023000 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(114))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 32)^{4} \)
|
| $3$ |
\( (T - 243)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 29\!\cdots\!92 \)
|
| $11$ |
\( T^{4} + \cdots + 35\!\cdots\!60 \)
|
| $13$ |
\( T^{4} + \cdots - 18\!\cdots\!96 \)
|
| $17$ |
\( T^{4} + \cdots + 58\!\cdots\!00 \)
|
| $19$ |
\( (T - 2476099)^{4} \)
|
| $23$ |
\( T^{4} + \cdots - 28\!\cdots\!40 \)
|
| $29$ |
\( T^{4} + \cdots + 32\!\cdots\!68 \)
|
| $31$ |
\( T^{4} + \cdots - 15\!\cdots\!92 \)
|
| $37$ |
\( T^{4} + \cdots - 77\!\cdots\!96 \)
|
| $41$ |
\( T^{4} + \cdots - 60\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 49\!\cdots\!20 \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!60 \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots - 22\!\cdots\!72 \)
|
| $61$ |
\( T^{4} + \cdots - 13\!\cdots\!64 \)
|
| $67$ |
\( T^{4} + \cdots - 52\!\cdots\!80 \)
|
| $71$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 50\!\cdots\!64 \)
|
| $79$ |
\( T^{4} + \cdots + 40\!\cdots\!92 \)
|
| $83$ |
\( T^{4} + \cdots - 11\!\cdots\!60 \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 97\!\cdots\!00 \)
|
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