Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,8,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, 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("441.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.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: | \(137.761796238\) |
| 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: |
\( x^{4} - x^{3} - 306x^{2} - 228x + 10152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{4} - x^{3} - 306x^{2} - 228x + 10152 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 236\nu + 116 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 2\nu - 153 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta _1 + 153 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 8\beta_{2} + 242\beta _1 + 343 \)
|
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 |
|
−15.2332 | 0 | 104.050 | −245.134 | 0 | 0 | 364.830 | 0 | 3734.18 | ||||||||||||||||||||||||||||||
| 1.2 | −6.80218 | 0 | −81.7304 | 12.5610 | 0 | 0 | 1426.62 | 0 | −85.4422 | |||||||||||||||||||||||||||||||
| 1.3 | 5.62848 | 0 | −96.3202 | 502.942 | 0 | 0 | −1262.58 | 0 | 2830.80 | |||||||||||||||||||||||||||||||
| 1.4 | 17.4069 | 0 | 175.000 | −74.3690 | 0 | 0 | 818.128 | 0 | −1294.53 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 441.8.a.r | 4 | |
| 3.b | odd | 2 | 1 | 147.8.a.h | 4 | ||
| 7.b | odd | 2 | 1 | 441.8.a.q | 4 | ||
| 7.d | odd | 6 | 2 | 63.8.e.c | 8 | ||
| 21.c | even | 2 | 1 | 147.8.a.i | 4 | ||
| 21.g | even | 6 | 2 | 21.8.e.a | ✓ | 8 | |
| 21.h | odd | 6 | 2 | 147.8.e.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.e.a | ✓ | 8 | 21.g | even | 6 | 2 | |
| 63.8.e.c | 8 | 7.d | odd | 6 | 2 | ||
| 147.8.a.h | 4 | 3.b | odd | 2 | 1 | ||
| 147.8.a.i | 4 | 21.c | even | 2 | 1 | ||
| 147.8.e.k | 8 | 21.h | odd | 6 | 2 | ||
| 441.8.a.q | 4 | 7.b | odd | 2 | 1 | ||
| 441.8.a.r | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{4} - T_{2}^{3} - 306T_{2}^{2} - 228T_{2} + 10152 \)
|
|
\( T_{5}^{4} - 196T_{5}^{3} - 140157T_{5}^{2} - 7379364T_{5} + 115169580 \)
|
|
\( T_{13}^{4} - 3794T_{13}^{3} - 148017327T_{13}^{2} + 541543052948T_{13} + 3278795227094912 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 10152 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 196 T^{3} + \cdots + 115169580 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 307523289329820 \)
|
| $13$ |
\( T^{4} + \cdots + 32\!\cdots\!12 \)
|
| $17$ |
\( T^{4} + \cdots - 25\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots - 84\!\cdots\!56 \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots - 65\!\cdots\!92 \)
|
| $31$ |
\( T^{4} + \cdots - 63\!\cdots\!49 \)
|
| $37$ |
\( T^{4} + \cdots - 63\!\cdots\!68 \)
|
| $41$ |
\( T^{4} + \cdots - 81\!\cdots\!40 \)
|
| $43$ |
\( T^{4} + \cdots + 27\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 17\!\cdots\!80 \)
|
| $53$ |
\( T^{4} + \cdots - 80\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots - 24\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 18\!\cdots\!20 \)
|
| $71$ |
\( T^{4} + \cdots - 21\!\cdots\!92 \)
|
| $73$ |
\( T^{4} + \cdots + 18\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots - 18\!\cdots\!69 \)
|
| $83$ |
\( T^{4} + \cdots - 20\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 15\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 98\!\cdots\!56 \)
|
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